Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Nonlinear tensor product approximation Vladimir Temlyakov ICERM; October 3, 2014 Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Introduction 1 Best nonlinear approximation 2 Constructive nonlinear approximation 3 Lemmas for trigonometric polynomials 4 Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Best multilinear approximation We are interested in approximation of a multivariate function f ( x 1 , . . . , x d ) by linear combinations of products u 1 ( x 1 ) · · · u d ( x d ) of univariate functions u i ( x i ), i = 1 , . . . , d . Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Best multilinear approximation We are interested in approximation of a multivariate function f ( x 1 , . . . , x d ) by linear combinations of products u 1 ( x 1 ) · · · u d ( x d ) of univariate functions u i ( x i ), i = 1 , . . . , d . Definition For a function f ( x 1 , . . . , x d ) denote M d � � u i Θ M ( f ) X := j } , j =1 ,..., M , i =1 ,..., d � f ( x 1 , . . . , x d ) − inf j ( x i ) � X { u i j =1 i =1 and for a function class F define Θ M ( F ) X := sup Θ M ( f ) X . f ∈ F Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Best multilinear approximation We are interested in approximation of a multivariate function f ( x 1 , . . . , x d ) by linear combinations of products u 1 ( x 1 ) · · · u d ( x d ) of univariate functions u i ( x i ), i = 1 , . . . , d . Definition For a function f ( x 1 , . . . , x d ) denote M d � � u i Θ M ( f ) X := j } , j =1 ,..., M , i =1 ,..., d � f ( x 1 , . . . , x d ) − inf j ( x i ) � X { u i j =1 i =1 and for a function class F define Θ M ( F ) X := sup Θ M ( f ) X . f ∈ F In the case X = L p we write p instead of L p in the notation. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Multilinear dictionary In other words we are interested in studying M -term approximations of functions with respect to the dictionary d Π d := { g ( x 1 , . . . , x d ) : g ( x 1 , . . . , x d ) = � u i ( x i ) } i =1 where u i ( x i ) are arbitrary univariate functions. We discuss the case of 2 π -periodic functions of d variables and approximate them in p the normalized in L p dictionary Π d of the L p spaces. Denote by Π d 2 π -periodic functions. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Mega problem Problem Find a simple algorithm such that for any f ∈ L p it provides after M ≤ m φ ( m ) iterations an M-term with respect to Π d approximant A M ( f ) such that � f − A M ( f ) � p ≤ C 1 Θ m ( f ) p . Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Mega problem Problem Find a simple algorithm such that for any f ∈ L p it provides after M ≤ m φ ( m ) iterations an M-term with respect to Π d approximant A M ( f ) such that � f − A M ( f ) � p ≤ C 1 Θ m ( f ) p . Simple – incremental, greedy. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Mega problem Problem Find a simple algorithm such that for any f ∈ L p it provides after M ≤ m φ ( m ) iterations an M-term with respect to Π d approximant A M ( f ) such that � f − A M ( f ) � p ≤ C 1 Θ m ( f ) p . Simple – incremental, greedy. Ideally, φ ( m ) = C 2 . Otherwise, the slower growing φ ( m ) the better. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials What does give a hope? In the case d = 2, p = 2 the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm (Orthogonal Matching Pursuit) solve the problem with φ ( m ) = 1, C 1 = 1. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials What does give a hope? In the case d = 2, p = 2 the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm (Orthogonal Matching Pursuit) solve the problem with φ ( m ) = 1, C 1 = 1. For the trigonometric system we have. Theorem (T., 2014) Let D be the normalized in L p , 2 ≤ p < ∞ , real d-variate trigonometric system. Then for any f ∈ L p the Weak Chebyshev Greedy Algorithm with weakness parameter t gives � f C ( t , p , d ) m ln( m +1) � p ≤ C σ m ( f , D ) p . (1) Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Dictionary Definition A set of functions D from a Banach space X is a dictionary if each g ∈ D has norm one ( � g � := � g � X = 1) and the closure of span D coincides with X . Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Dictionary Definition A set of functions D from a Banach space X is a dictionary if each g ∈ D has norm one ( � g � := � g � X = 1) and the closure of span D coincides with X . Definition For a nonzero element f ∈ X we denote by F f a norming (peak) functional for f : � F f � = 1 , F f ( f ) = � f � . Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Dictionary Definition A set of functions D from a Banach space X is a dictionary if each g ∈ D has norm one ( � g � := � g � X = 1) and the closure of span D coincides with X . Definition For a nonzero element f ∈ X we denote by F f a norming (peak) functional for f : � F f � = 1 , F f ( f ) = � f � . The existence of such a functional is guaranteed by the Hahn-Banach theorem. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Chebyshev greedy algorithm Weak Chebyshev Greedy Algorithm (WCGA)( t ) Let t ∈ (0 , 1]. For a given f 0 we inductively define for each m ≥ 1 ϕ m ∈ D is any satisfying | F f m − 1 ( ϕ m ) | ≥ t sup | F f m − 1 ( g ) | . g ∈D Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Chebyshev greedy algorithm Weak Chebyshev Greedy Algorithm (WCGA)( t ) Let t ∈ (0 , 1]. For a given f 0 we inductively define for each m ≥ 1 ϕ m ∈ D is any satisfying | F f m − 1 ( ϕ m ) | ≥ t sup | F f m − 1 ( g ) | . g ∈D Define Φ m := span { ϕ j } m j =1 , and define G m to be the best approximant to f 0 from Φ m . Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Chebyshev greedy algorithm Weak Chebyshev Greedy Algorithm (WCGA)( t ) Let t ∈ (0 , 1]. For a given f 0 we inductively define for each m ≥ 1 ϕ m ∈ D is any satisfying | F f m − 1 ( ϕ m ) | ≥ t sup | F f m − 1 ( g ) | . g ∈D Define Φ m := span { ϕ j } m j =1 , and define G m to be the best approximant to f 0 from Φ m . Denote f m := f − G m . Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Best m -term approximation Definition We let Σ m ( D ) denote the collection of all functions (elements) in X which can be expressed as a linear combination of at most m elements of D . Thus each function f ∈ Σ m ( D ) can be written in the form � f = c g g , Λ ⊂ D , #Λ ≤ m , g ∈ Λ where the c g are real or complex numbers. Vladimir Temlyakov Nonlinear tensor product approximation
Introduction Best nonlinear approximation Constructive nonlinear approximation Lemmas for trigonometric polynomials Best m -term approximation Definition We let Σ m ( D ) denote the collection of all functions (elements) in X which can be expressed as a linear combination of at most m elements of D . Thus each function f ∈ Σ m ( D ) can be written in the form � f = c g g , Λ ⊂ D , #Λ ≤ m , g ∈ Λ where the c g are real or complex numbers. Definition For a function f ∈ X we define its best m -term approximation error σ m ( f ) := σ m ( f , D ) := a ∈ Σ m ( D ) � f − a � . inf Vladimir Temlyakov Nonlinear tensor product approximation
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