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[null,"1.","Record Nr.",null,"UNINA9910865238103321"]
[null,null,"Autore",null,"Noonan Jack"]
[null,null,"Titolo",null,"High-Dimensional Optimization : Set Exploration in the Non-Asymptotic Regime"]
[null,null,"Pubbl/distr/stampa",null,"Cham : , : Springer International Publishing AG, , 2024"]
[{"$nulls":4},"©2024"]
[null,null,"ISBN",null,"9783031589096"]
[{"$nulls":4},"9783031589089"]
[null,null,"Edizione",null,"[1st ed.]"]
[null,null,"Descrizione fisica",null,"1 online resource (153 pages)"]
[null,null,"Collana",null,"SpringerBriefs in Optimization Series"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,"Nota di contenuto",null,"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-"]
[{"$nulls":4},"Based Approximations -- Approximation for One Ball -- Approximation for n Balls -- Simulation Study -- 2.2 Construction of Efficient Exploration Schemes -- 2.2.1 The Probability of Covering as a Function of α and δ."]
[{"$nulls":4},"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."]