Outline High Dimensional Approxima- tion - Background and Sources Dahmen High Dimensional Approximation - Outline Background and Sources Wolfgang Dahmen Seminar: USC, High Dimensional Approximation, Feb 13, 2008
Outline High Outline Dimensional Approxima- tion - Background Learning Theory 1 and Sources Regression Dahmen Basic Concepts Outline Remedies Specific Problem Areas 2 Climatology- An Example Finance Electronic Structure Calculation Stochastic Multiscale Modeling An Instance of “Manifold Learning” Methodological Aspects 3 Summary of Key Issues Compressed Sensing Greedy Techniques
Learning Theory Specific Problem Areas Methodological Aspects Regression High Learning Theory - Regression Problem Dimensional Approxima- tion - ρ unknown measure on Z := X × Y Background and Sources � Dahmen f ρ ( x ) := yd ρ ( y | x ) = E ( y | x ) Learning Y Theory Regression Basic Concepts Remedies Y Specific Problem Areas Climatology- An Example Finance Electronic Structure Calculation Stochastic Multiscale Modeling An Instance of “Manifold Learning” x x’ X Methodological Aspects � ( y − f ( x )) 2 d ρ � E ( f ) = E ( f ρ ) + � f − f ρ � 2 E ( f ) := Summary of Key L 2 ( X ,ρ X ) Issues Compressed Z Sensing
Learning Theory Specific Problem Areas Methodological Aspects Basic Concepts High Concepts - Obstructions Dimensional Approxima- tion - Background and Sources Dahmen Relevant concepts Learning Nonparametric estimation, concentration inequalities, Theory nonlinear approximation Regression Basic Concepts Remedies Specific Problem Solution strategies Areas Climatology- An Example Adaptive partitioning Finance Electronic Structure Complexity regularization (model selection) Calculation Stochastic Multiscale Modeling An Instance of “Manifold dim X large - Curse of dimensionality: Learning” Are there ways around it? Methodological Aspects Summary of Key Issues Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Remedies High Ameliorating the curse of dimensionality Dimensional Approxima- tion - Background Dimensionwise decompositions - ANOVA-type schemes and Sources Dahmen Kernel methods, neural networks Learning Sparse grids, hyperbolic cross approximation Theory Regression Kronnecker-product approximation Basic Concepts Remedies Dimension reduction - “learning” embedded manifolds Specific Problem Recovery schemes: Areas Climatology- An Example Greedy algorithms Finance Electronic Procedural recovery (sparse occupancy trees, Sprecher’s Structure Calculation alg.) Stochastic Multiscale Modeling A higher level of difficulty: An Instance of “Manifold Learning” Learning on Banach spaces ( dim X = ∞ ) Methodological Aspects Learning implicitly given functions - e.g. solutions of Summary of Key Issues stochastic PDEs Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Climatology- An Example High Dynamical System Input Dimensional Approxima- tion - Background and Sources ∂ψ ∂ t + D ( ψ, x ) = P ( ψ, x ) Dahmen Learning Theory ψ 3D prognostic dependent variable, e.g. temperature, Regression Basic Concepts pressure, moisture, etc. Remedies Specific x 3D dependent variable, e.g. latitude, longitude, Problem Areas height, Climatology- An Example D model dynamics, PDE of motion, thermodynamcs, Finance Electronic Structure balance laws, etc. Calculation Stochastic Multiscale P model physics, long, short range athmospheric Modeling An Instance of radiation , turbulence, convection, clouds, interactions “Manifold Learning” with land, chemistry, etc. so complicated even as Methodological Aspects simplified parametrized versions – based on solving Summary of Key Issues deterministic equations Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Climatology- An Example High Alternative: Learning Dimensional Approxima- tion - Background and Sources Dahmen Instead of computing the forcing terms by solving Learning deterministic equations, taking most of the time, one tries to Theory Regression “learn” P from aquired data Basic Concepts Remedies Problem: Specific Given Z = { z i = ( x i , y i ) ∈ X × Y ⊂ R d × d ′ : i = 1 , . . . , N } Problem Areas find f : R d → R d ′ with f ( x i ) = y i , i = 1 , . . . , N Climatology- An Example Finance Electronic Structure Possible strategy: Sparse occupancy trees Calculation Stochastic Multiscale Question: reasonable error bounds?- concentration of Modeling An Instance of “Manifold measure phenomenon Learning” Methodological Aspects Summary of Key Issues Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Finance High Finance - high dimensional integration Dimensional Approxima- tion - Background and Sources Dahmen In the US mortgages last 30 years and may be repaid each Learning month, which gives 12 × 30 = 360 repayment Theory Regression possibilities � Basic Concepts Remedies Computation of 360-dimensional expected value Specific Problem 1 1 Areas � � Climatology- An Example · · · f ( x 1 , . . . , x 360 ) dx 1 · · · dx 360 Finance Electronic Structure 0 0 Calculation Stochastic Multiscale Note: Quadrature rule with k nodes in [ 0 , 1 ] requires k 360 Modeling An Instance of “Manifold point evaluations... Learning” Methodological Aspects Summary of Key Issues Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Electronic Structure Calculation High Electronic Structure Calculation Dimensional Approxima- tion - Background and Sources Dahmen Learning Theory Goal: Numerical simulation of molecular phenomena in Regression Basic Concepts chemistry, molecular biology, semiconductor devices, Remedies material sciences... Specific Problem Areas “Ab-Initio” Calculations based on first principles in quantum Climatology- An Example Finance mechanics (ignoring relativistic effects and using the Electronic Structure Born-Oppenheimer approximate Model) � Calculation Stochastic Multiscale Modeling An Instance of “Manifold Learning” Methodological Aspects Summary of Key Issues Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Electronic Structure Calculation High Quantum mechanical postulates: Dimensional Approxima- tion - Background and Sources System of N identicle (non-relativistic) particles with spin s i Dahmen described by a state function ψ : R 3 N ⊗ S N → C , Learning ψ ( x 1 , s 1 ; . . . ; x N , s N ) , � ψ, ψ � = 1 Theory Regression Basic Concepts Remedies Specific ψ satisfies (stat.) Schr¨ odinger equation with Hamiltonian H Problem Areas H ψ = E 0 ψ, E 0 = � ψ,ψ � = 1 � H ψ, ψ � min Climatology- An Example Finance Electronic Structure Born-Oppenheim: Calculation Stochastic Multiscale Modeling N M � z j � − 1 | x i − R j | + 1 1 � � � An Instance of H = 2 ∆ i − “Manifold Learning” 2 | x i − x j | i = 1 j = 1 j � = i Methodological Aspects Summary of Key z j = charge of j th nucleus at position R j Issues Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Stochastic Multiscale Modeling High Typical Applications Dimensional Approxima- tion - Background and Sources Dahmen Simulation of porous media flow, contamination prediction, well protection Learning Theory Regression Understanding heterogeneous materials like concrete Basic Concepts Remedies Classical diffusion equation: Specific Problem Areas D ⊂ R d , − div ( A ∇ u ) = f in u | ∂ D = 0 , ( d = 2 , 3 ) (1) Climatology- An Example Finance Electronic A = A ( x ) describes diffusivity of the material Structure Calculation Stochastic Multiscale Problem: In heterogeneous porous media the small scales Modeling An Instance of “Manifold of the material make it impossible to describe all details by Learning” A and to resolve them numerically Methodological Aspects Summary of Key Issues Compressed Sensing
Learning Theory Specific Problem Areas Methodological Aspects Stochastic Multiscale Modeling High Stochastic Model Dimensional Approxima- tion - Idea: view A as a random field ( A = aI scalar) about which Background and Sources (coarsely sampled) measurements provide uncertain Dahmen information: Learning Theory a = a ( · , ω ) : ω → L ∞ ( D ) =: X , ω ∈ Ω Regression Basic Concepts Remedies Specific where Problem Areas Climatology- An (Ω , Σ , ρ ) probability space on data space X Example Finance Electronic Structure Calculation Stochastic Multiscale Modeling Proposition: An Instance of “Manifold When a ( · , ω ) stays bounded away from zero ρ -a.s. then (1) Learning” Methodological is well posed, i.e. there exists a unique u ( · , ω ) : Ω → H 1 0 ( D ) Aspects Summary of Key which is a weak solution of (1). Issues Compressed Sensing
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