Data driven regularization by projection Andrea Aspri Joint work with Y. Korolev and O. Scherzer Joint meeting Fudan University and RICAM Shanghai & Linz - 10, June 2020 logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Motivation Given Au = y and y δ noisy measurements s.t. � y − y δ � ≤ δ . Goal: Given measurements y δ , reconstruct the unknown quantity u . Assume we have { u i , y i } i =1 , ··· , n s.t. Au i = y i Questions How can we use these pairs in the reconstruction process? How can we use these pairs when A is not explicitly known ? logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Learning an operator: is it possible? Main goal: develop stable algorithms for finding u such that Au = y without explicit knowledge of A , but having only training pairs : { u i , y i } i =1 , ··· , n s.t. Au i = y i noisy measurements y δ , s.t. � y − y δ � ≤ δ Question Is there a regularization method capable of learning a linear operator? Spoiler: Yes, there is... logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Learning an operator: is it possible? Main goal: develop stable algorithms for finding u such that Au = y without explicit knowledge of A , but having only training pairs : { u i , y i } i =1 , ··· , n s.t. Au i = y i noisy measurements y δ , s.t. � y − y δ � ≤ δ Question Is there a regularization method capable of learning a linear operator? Spoiler: Yes, there is... logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Terminology & Notation u i are “training images” y i are “training data” U n := Span { u i } i =1 , ··· , n Y n := Span { y i } i =1 , ··· , n P U n orthogonal projection onto U n P Y n orthogonal projection onto Y n u † solution to Au = y . logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Main assumptions 1. Operator ◮ A : U → Y , with U , Y Hilbert spaces. ◮ A is bounded, linear and injective (but A − 1 is unbounded). 2. Data ◮ Linear independence of { u i } i =1 , ··· , n , ∀ n ∈ N . ◮ Uniform boundedness: ∃ C u > 0 s.t. � u i � ≤ C u , ∀ i ∈ N . ◮ Sequentiality: training pairs are nested, i.e. { u i , y i } i =1 , ··· , n +1 = { u i , y i } i =1 , ··· , n ∪ { u n +1 , y n +1 } , hence U n ⊂ U n +1 and Y n ⊂ Y n +1 , ∀ n ∈ N . ◮ Density: training images spaces are dense in U , i.e., � n ∈ N U n = U . Consequences: Training data y i are linearly independent and uniformly bounded as well; Training data spaces are dense in R ( A ). logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Main assumptions 1. Operator ◮ A : U → Y , with U , Y Hilbert spaces. ◮ A is bounded, linear and injective (but A − 1 is unbounded). 2. Data ◮ Linear independence of { u i } i =1 , ··· , n , ∀ n ∈ N . ◮ Uniform boundedness: ∃ C u > 0 s.t. � u i � ≤ C u , ∀ i ∈ N . ◮ Sequentiality: training pairs are nested, i.e. { u i , y i } i =1 , ··· , n +1 = { u i , y i } i =1 , ··· , n ∪ { u n +1 , y n +1 } , hence U n ⊂ U n +1 and Y n ⊂ Y n +1 , ∀ n ∈ N . ◮ Density: training images spaces are dense in U , i.e., � n ∈ N U n = U . Consequences: Training data y i are linearly independent and uniformly bounded as well; Training data spaces are dense in R ( A ). logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Main assumptions 1. Operator ◮ A : U → Y , with U , Y Hilbert spaces. ◮ A is bounded, linear and injective (but A − 1 is unbounded). 2. Data ◮ Linear independence of { u i } i =1 , ··· , n , ∀ n ∈ N . ◮ Uniform boundedness: ∃ C u > 0 s.t. � u i � ≤ C u , ∀ i ∈ N . ◮ Sequentiality: training pairs are nested, i.e. { u i , y i } i =1 , ··· , n +1 = { u i , y i } i =1 , ··· , n ∪ { u n +1 , y n +1 } , hence U n ⊂ U n +1 and Y n ⊂ Y n +1 , ∀ n ∈ N . ◮ Density: training images spaces are dense in U , i.e., � n ∈ N U n = U . Consequences: Training data y i are linearly independent and uniformly bounded as well; Training data spaces are dense in R ( A ). logo.png Andrea Aspri (RICAM) Data driven regularization by projection
Regularization by projection Approximate u † using the Minimum Norm Solution (MNS) of finite dimensional problems Least-Squares Proj. Dual Least-Squares Proj. (1) AP n u = y (2) Q n Au = Q n y P n = orthogonal projection onto a Q n orthogonal projection onto a finite dimensional space of U . finite dimensional space of Y . Our idea to use training pairs Choose P n = P U n . Choose Q n = P Y n It can be proven It can be proven u U n = A − 1 P Y n y u Y n = P A ∗ Y n u U MNS : MNS : n In general u U n � u † . In this case u Y n → u † . Engl, H. W. and Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Springer (1996). Seidman T. I., Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Opt. Th. logo.png Appl. (1980). Andrea Aspri (RICAM) Data driven regularization by projection
Regularization by projection Approximate u † using the Minimum Norm Solution (MNS) of finite dimensional problems Least-Squares Proj. Dual Least-Squares Proj. (1) AP n u = y (2) Q n Au = Q n y P n = orthogonal projection onto a Q n orthogonal projection onto a finite dimensional space of U . finite dimensional space of Y . Our idea to use training pairs Choose P n = P U n . Choose Q n = P Y n It can be proven It can be proven u U n = A − 1 P Y n y u Y n = P A ∗ Y n u U MNS : MNS : n In general u U n � u † . In this case u Y n → u † . Engl, H. W. and Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Springer (1996). Seidman T. I., Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Opt. Th. logo.png Appl. (1980). Andrea Aspri (RICAM) Data driven regularization by projection
Regularization by projection Approximate u † using the Minimum Norm Solution (MNS) of finite dimensional problems Least-Squares Proj. Dual Least-Squares Proj. (1) AP n u = y (2) Q n Au = Q n y P n = orthogonal projection onto a Q n orthogonal projection onto a finite dimensional space of U . finite dimensional space of Y . Our idea to use training pairs Choose P n = P U n . Choose Q n = P Y n It can be proven It can be proven u U n = A − 1 P Y n y u Y n = P A ∗ Y n u U MNS : MNS : n In general u U n � u † . In this case u Y n → u † . Engl, H. W. and Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Springer (1996). Seidman T. I., Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Opt. Th. logo.png Appl. (1980). Andrea Aspri (RICAM) Data driven regularization by projection
Regularization by projection Approximate u † using the Minimum Norm Solution (MNS) of finite dimensional problems Least-Squares Proj. Dual Least-Squares Proj. (1) AP n u = y (2) Q n Au = Q n y P n = orthogonal projection onto a Q n orthogonal projection onto a finite dimensional space of U . finite dimensional space of Y . Our idea to use training pairs Choose P n = P U n . Choose Q n = P Y n It can be proven It can be proven u U n = A − 1 P Y n y u Y n = P A ∗ Y n u U MNS : MNS : n In general u U n � u † . In this case u Y n → u † . Engl, H. W. and Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Springer (1996). Seidman T. I., Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Opt. Th. logo.png Appl. (1980). Andrea Aspri (RICAM) Data driven regularization by projection
Regularization by projection Approximate u † using the Minimum Norm Solution (MNS) of finite dimensional problems Least-Squares Proj. Dual Least-Squares Proj. (1) AP n u = y (2) Q n Au = Q n y P n = orthogonal projection onto a Q n orthogonal projection onto a finite dimensional space of U . finite dimensional space of Y . Our idea to use training pairs Choose P n = P U n . Choose Q n = P Y n It can be proven It can be proven u U n = A − 1 P Y n y u Y n = P A ∗ Y n u U MNS : MNS : n In general u U n � u † . In this case u Y n → u † . Engl, H. W. and Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Springer (1996). Seidman T. I., Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Opt. Th. logo.png Appl. (1980). Andrea Aspri (RICAM) Data driven regularization by projection
Regularization by projection Approximate u † using the Minimum Norm Solution (MNS) of finite dimensional problems Least-Squares Proj. Dual Least-Squares Proj. (1) AP n u = y (2) Q n Au = Q n y P n = orthogonal projection onto a Q n orthogonal projection onto a finite dimensional space of U . finite dimensional space of Y . Our idea to use training pairs Choose P n = P U n . Choose Q n = P Y n It can be proven It can be proven u U n = A − 1 P Y n y u Y n = P A ∗ Y n u U MNS : MNS : n In general u U n � u † . In this case u Y n → u † . Engl, H. W. and Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Springer (1996). Seidman T. I., Nonconvergence results for the application of least-squares estimation to ill-posed problems, J. Opt. Th. logo.png Appl. (1980). Andrea Aspri (RICAM) Data driven regularization by projection
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