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Table of Contents
Intro
Preface
Contents
Projection of a Point onto a Convex Set via Charged Balls Method
1 Introduction
2 Charged Balls Method and its Modification
3 Case of the Set with Nonsmooth Boundary
4 Numerical Experiments
5 Conclusion
References
Towards Optimal Sampling for Learning Sparse Approximations in High Dimensions
1 Introduction
1.1 Main Problem
1.2 Overview
1.3 Additional Contributions
1.4 Related Literature
1.5 Outline
2 Preliminaries
2.1 Notation
2.2 Problem and Key Questions
2.3 Examples
2.4 Multi-Index Sets
3 Sparse Approximation via (Weighted) Least Squares
3.1 Computation of the Least-Squares Approximation
3.2 Accuracy, Stability and Sample Complexity
3.3 Monte Carlo Sampling
3.4 Optimal Sampling
3.5 Practical Optimal Sampling via Discrete Measures
3.6 Numerical Examples
3.7 Proofs of Theorems 2 and 3
4 Sparse Approximation via 1-Minimization
4.1 Formulation
4.2 Accuracy, Stability and Sample Complexity
4.3 Monte Carlo Sampling
4.4 `Optimal' Sampling
4.5 `Optimal' Sampling and Discrete Measures
4.6 Further Discussion and Numerical Examples
4.7 Proof of Theorems 5 and 6
5 A Novel Approach for Sparse Polynomial Approximation on Irregular Domains
5.1 Method
5.2 Orderings
5.3 Numerical Examples
6 Structured Sparse Approximation
6.1 Weighted Sparsity and Weighted 1-Minimization
6.2 Sparsity in Lower Sets
6.3 Sampling and Numerical Experiments
7 Conclusions and Challenges
References
Recent Theoretical Advances in Non-Convex Optimization
1 Introduction
2 Preliminaries
2.1 Global Optimization Is NP-Hard
2.2 Lower Complexity Bound for Global Optimization
2.3 Examples of Non-Convex Problems
2.3.1 Problems with Hidden Convexity or Analytic Solutions
2.3.2 Problems with Convergence Results
2.3.3 Geometry of Non-Convex Optimization Problems
3 Deterministic First-Order Methods
3.1 Unconstrained Minimization
3.2 Incorporating Simple Constraints
3.3 Incorporating Momentum for Acceleration
4 Stochastic First-Order Methods
4.1 General View on Optimal Deterministic and Stochastic First-Order Methods for Non-Convex Optimization
4.1.1 Deterministic Case
4.1.2 Stochastic Case: Uniformly Bounded Variance
4.1.3 Stochastic Case: Finite-Sum Minimization
Preface
Contents
Projection of a Point onto a Convex Set via Charged Balls Method
1 Introduction
2 Charged Balls Method and its Modification
3 Case of the Set with Nonsmooth Boundary
4 Numerical Experiments
5 Conclusion
References
Towards Optimal Sampling for Learning Sparse Approximations in High Dimensions
1 Introduction
1.1 Main Problem
1.2 Overview
1.3 Additional Contributions
1.4 Related Literature
1.5 Outline
2 Preliminaries
2.1 Notation
2.2 Problem and Key Questions
2.3 Examples
2.4 Multi-Index Sets
3 Sparse Approximation via (Weighted) Least Squares
3.1 Computation of the Least-Squares Approximation
3.2 Accuracy, Stability and Sample Complexity
3.3 Monte Carlo Sampling
3.4 Optimal Sampling
3.5 Practical Optimal Sampling via Discrete Measures
3.6 Numerical Examples
3.7 Proofs of Theorems 2 and 3
4 Sparse Approximation via 1-Minimization
4.1 Formulation
4.2 Accuracy, Stability and Sample Complexity
4.3 Monte Carlo Sampling
4.4 `Optimal' Sampling
4.5 `Optimal' Sampling and Discrete Measures
4.6 Further Discussion and Numerical Examples
4.7 Proof of Theorems 5 and 6
5 A Novel Approach for Sparse Polynomial Approximation on Irregular Domains
5.1 Method
5.2 Orderings
5.3 Numerical Examples
6 Structured Sparse Approximation
6.1 Weighted Sparsity and Weighted 1-Minimization
6.2 Sparsity in Lower Sets
6.3 Sampling and Numerical Experiments
7 Conclusions and Challenges
References
Recent Theoretical Advances in Non-Convex Optimization
1 Introduction
2 Preliminaries
2.1 Global Optimization Is NP-Hard
2.2 Lower Complexity Bound for Global Optimization
2.3 Examples of Non-Convex Problems
2.3.1 Problems with Hidden Convexity or Analytic Solutions
2.3.2 Problems with Convergence Results
2.3.3 Geometry of Non-Convex Optimization Problems
3 Deterministic First-Order Methods
3.1 Unconstrained Minimization
3.2 Incorporating Simple Constraints
3.3 Incorporating Momentum for Acceleration
4 Stochastic First-Order Methods
4.1 General View on Optimal Deterministic and Stochastic First-Order Methods for Non-Convex Optimization
4.1.1 Deterministic Case
4.1.2 Stochastic Case: Uniformly Bounded Variance
4.1.3 Stochastic Case: Finite-Sum Minimization