Application of K ahler manifold to signal processing and Bayesian - PowerPoint PPT Presentation

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion Application of K¨ ahler manifold to signal processing and Bayesian inference Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 1. Department of Applied Mathematics and Statistics 2. Department of Physics and Astronomy SUNY at Stony Brook September 23, 2014 MaxEnt 2014, Amboise Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion Table of contents 1 Review on K¨ ahler manifold 2 K¨ ahlerian information geometry for signal processing 3 Geometric shrinkage priors 4 Example: ARFIMA 5 Conclusion Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold and information geometry Implications of K¨ ahler manifold differential geometry, algebraic geometry Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold and information geometry Implications of K¨ ahler manifold differential geometry, algebraic geometry superstring theory and supergravity in theoretical physics Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold and information geometry Implications of K¨ ahler manifold differential geometry, algebraic geometry superstring theory and supergravity in theoretical physics Information Geometry Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold and information geometry Implications of K¨ ahler manifold differential geometry, algebraic geometry superstring theory and supergravity in theoretical physics Information Geometry Barbaresco (2006, 2012, 2014): K¨ ahler manifold and Koszul information geometry Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold and information geometry Implications of K¨ ahler manifold differential geometry, algebraic geometry superstring theory and supergravity in theoretical physics Information Geometry Barbaresco (2006, 2012, 2014): K¨ ahler manifold and Koszul information geometry Zhang and Li (2013): symplectic and K¨ ahler structures in divergence function Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold Definition The K¨ ahler manifold is the Hermitian manifold with the closed K¨ ahler two-form. In the metric expression, g ij = g ¯ j = 0 i ¯ ∂ i g j ¯ k = ∂ j g i ¯ k = 0 Any advantages? Let’s discuss later. Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion Linear systems and information geometry Linear systems are described by the transfer function h ( w ; ξ ) y ( w ) = h ( w ; ξ ) x ( w ; ξ ) where input x and output y . The metric tensor for the filter � π g µν ( ξ ) = 1 ( ∂ µ log S )( ∂ ν log S ) dw 2 π − π where S ( w ; ξ ) = | h ( w ; ξ ) | 2 . Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion z-transformation h ( z ; ξ ) = � ∞ r =0 h r ( ξ ) z − r ∞ ∞ h r � z − r ) = log h 0 + � η r z − r log h ( z ; ξ ) = log h 0 + log (1 + h 0 r =1 r =1 The metric tensor in terms of transfer function 1 � � dz log h + log ¯ log h + log ¯ � � � g µν = ∂ µ h ∂ ν h 2 π i z | z | =1 where µ, ν run holomorphic and anti-holomorphic indices. Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion The metric tensors in holomorphic and anti-holomorphic coordinates 1 � ∂ i log h ( z ; ξ ) ∂ j log h ( z ; ξ ) dz g ij ( ξ ) = 2 π i z | z | =1 1 � ξ ) dz j log ¯ z ; ¯ j ( ξ ) = ∂ i log h ( z ; ξ ) ∂ ¯ h (¯ g i ¯ 2 π i z | z | =1 The metric tensor g ij = ∂ i log h 0 ∂ j log h 0 ∞ j log ¯ � j = ∂ i log h 0 ∂ ¯ h 0 + ∂ i η r ∂ ¯ j ¯ η r g i ¯ r =1 Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler manifold for signal processing Theorem Given a holomorphic transfer function h ( z ; ξ ), the information geometry of a signal processing model is K¨ ahler manifold if and only if h 0 is a constant in ξ . ( ⇒ ) If the geometry is K¨ ahler, it should be Hermitian imposing g ij = ∂ i log ( h 0 ) ∂ j log ( h 0 ) = 0 → h 0 constant in ξ ( ⇐ ) If h 0 is a constant in ξ , the metric tensor is given in ∞ � g ij = 0 and g i ¯ j = ∂ i η r ∂ ¯ j ¯ η r → Hermitian r =1 j d ξ i ∧ d ¯ ξ j The K¨ ahler two-form is closed : Ω = ig i ¯ Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion K¨ ahler potential for signal processing On the K¨ ahler manifold, the metric tensor is j = ∂ i ∂ ¯ j K g i ¯ where the K¨ ahler potential K . Corollary Given K¨ ahler geometry, the K¨ ahler potential of the geometry is the square of the Hardy norm of the log-transfer function. 1 � � ∗ dz � �� K = log h ( z ; ξ ) log h ( z ; ξ ) 2 π i z | z | =1 = || log h ( z ; ξ ) || 2 H 2 Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion Benefits of K¨ ahlerian information geometry 1. Calculation of geometric objects is simplified. g i ¯ j = ∂ i ∂ ¯ j K , Γ ij , ¯ k = ∂ i ∂ j ∂ ¯ k K R i m Γ i mn = ∂ ¯ jn , R i ¯ j = − ∂ i ∂ ¯ j log G j ¯ 2. Easy α -generalization and linear order correction in α Γ ( α ) = Γ + α T , R ( α ) = R + α∂ T 3. Submanifolds of K¨ ahler is K¨ ahler. 4. Laplace-Beltrami operator:∆ = 2 g i ¯ j ∂ i ∂ ¯ j Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian

Review on K¨ ahler manifold K¨ ahlerian information geometry for signal processing Geometric shrinkage priors Example: ARFIMA Conclusion Komaki’s shrinkage prior for Bayesian inference Komaki (2006): The difference in risk functions is given by E ( D KL ( p ( y | ξ ) || p π J ( y | x ( N ) )) | ξ )) − E ( D KL ( p ( y | ξ ) || p π I ( y | x ( N ) )) | ξ )) � π I � π I � π I 1 − 1 π J � � � 2 N 2 g ij ∂ i log + o ( N − 2 ) = ∂ j log ∆ N 2 π J π J π I π J If ψ = π I /π J is superharmonic, p π I outperforms p π J . Superharmonic prior π I , Jeffreys prior π J Superharmonicity of functions is hard to check. In particular, in high-dimensional curved geometry! Jaehyung Choi ∗ , 1 , 2 Andrew P. Mullhaupt 1 Application of K¨ ahler manifold to signal processing and Bayesian