High-Dimensional Optimization : Set Exploration in the Non-Asymptotic Regime

• Autore: Noonan Jack
• Pubblicazione: Cham : , : Springer International Publishing AG, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (153 pages)
• Altri autori: ZhigljavskyAnatoly
• Nota di contenuto: Intro -- Introduction -- Contents -- Notation and Abbreviations -- 1 High-Dimensional Cubes, Balls and Spherically Symmetric Distributions -- 1.1 Spherically Symmetric and Beta Distributions -- 1.1.1 Beta-Distributed Random Variables -- 1.1.2 Uniform Distribution on a Sphere -- 1.1.3 Spherically Symmetric Random Vectors -- 1.1.4 Ball and Sphere -- 1.2 High-Dimensional Cube -- 1.2.1 Concentration of Mass in the Cube -- 1.2.2 Approximations for the Distribution of "026B30D U"026B30D 2 -- 1.3 The Squared Distance "026B30D X-Y"026B30D 2 When X Either Is Spherically Symmetric or Has i.i.d. Symmetric Components -- 1.3.1 Distribution of the Squared Distance When X Is Spherically Symmetric -- 1.3.2 The First Two Moments of "026B30D X-Y"026B30D 2 -- 1.3.3 The Third Moment of "026B30D X-Y"026B30D 2 -- 1.3.4 The Fourth Moment of "026B30D X-Y"026B30D 2 -- 1.4 Approximation of the Volume of Intersection of a Ball and a Cube -- 1.4.1 Variability in the Volumes of Intersection -- 1.4.2 Approximations with Spherically Symmetric Models -- 1.4.3 CLT-Based Approximations -- 1.4.4 Numerical Comparison of CLT-Based Approximations -- 1.4.5 Comparison of Approximations of Different Origins -- References -- 2 Space Exploration -- 2.1 Volume of Intersection of a Cube and n Balls -- 2.1.1 Covering and Partial Covering -- 2.1.2 Partial Covering: Asymptotic Considerations -- 2.1.3 Asymptotic Versus Non-asymptotic Regimes -- 2.1.4 The Family of Random Designs Considered -- 2.1.5 Approximations with Spherically Symmetric Models -- Approximations with One Ball -- Simulation Study -- 2.1.6 Approximations for n Balls -- Simulation Study -- 2.1.7 CLT-Based Approximations -- Approximation for One Ball -- Approximation for n Balls -- Simulation Study -- 2.2 Construction of Efficient Exploration Schemes -- 2.2.1 The Probability of Covering as a Function of α and δ. ; 2.2.2 Efficiency Plots -- 2.2.3 Practical Recommendations -- 2.3 Quantization -- 2.3.1 Bounds for Optimal Quantizers -- 2.3.2 Boundary Correction for Nearest Neighbor Distances -- 2.3.3 Approximating Quantization Error for Finite n -- Approximating Quantization Using Partial Covering -- Approximations Based on the Use of the Spherical Model -- Simulation Study -- CLT-Based Approximations for the Quantization Error -- Simulation Study -- Approximating Mean Squared Quantization Error Using Extreme Value Theory -- A Simple Approximation for Mean Squared Quantization Error -- Simulation Study -- 2.3.4 Efficient Exploration Designs for Quantization -- 2.3.5 Equivalence to the Problem of Partial Covering -- 2.4 Quantization Using the Checkerboard Lattice Points -- 2.4.1 Reformulation in Terms of the Voronoi Cells -- Re-normalization of the Quantization Error -- Voronoi Cells for Dn,δ -- 2.4.2 Explicit Formulae for the Quantization Error -- 2.4.3 Closed-Form Expressions for the Coverage Area -- Reduction to Voronoi Cells -- Expressing Fd(Dn,δ,r) Through Fd,Z(r) -- Simple Bounds for Fd(Dn,δ,r) -- Radius Required for Partial Covering Is Much Smaller than the Covering Radius -- Numerical Studies -- Quantization and Weak Covering Comparisons -- Accuracy of Covering Approximation and Dependence on δ -- Stochastic Dominance -- 2.4.4 The Checkerboard Lattice with Point at Zero -- An Auxiliary Result -- Normalised Mean Squared Quantization Error for Dn,δ,0(0) -- Quantization Error for the Design Dn,δ,0 -- Numerical Studies -- References.
• Titolo autorizzato: High-Dimensional Optimization
• ISBN: 9783031589096; 9783031589089
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910865238103321
• Lo trovi qui: Univ. Federico II
• Serie: SpringerBriefs in Optimization Series