Statistical Inference via Convex Optimization

Name: Statistical Inference via Convex Optimization
Brand: Princeton University Press
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Anatoli Juditsky Arkadi Nemirovski(Autor*in)

Princeton University Press

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Erschienen am 7. April 2020

656 Seiten

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Inhalt

Cover
Title
Copyright
Contents
List of Figures
Preface
Acknowledgements
Notational Conventions
About Proofs
On Computational Tractability
1 Sparse Recovery via l1 Minimization
1.1 Compressed Sensing: What is it about?
1.1.1 Signal Recovery Problem
1.1.2 Signal Recovery: Parametric and nonparametric cases
1.1.3 Compressed Sensing via l1 minimization: Motivation
1.2 Validity of sparse signal recovery via l1 minimization
1.2.1 Validity of l1 minimization in the noiseless case
1.2.2 Imperfect l1 minimization
1.2.3 Regular l1 recovery
1.2.4 Penalized l1 recovery
1.2.5 Discussion
1.3 Verifiability and tractability issues
1.3.1 Restricted Isometry Property and s-goodness of random matrices
1.3.2 Verifiable sufficient conditions for Qq(s, k)
1.3.3 Tractability of Q8(s, k)
1.4 Exercises for Chapter 1
1.5 Proofs
1.5.1 Proofs of Theorem 1.3, 1.4
1.5.2 Proof of Theorem 1.5
1.5.3 Proof of Proposition 1.7
1.5.4 Proof of Propositions 1.8 and 1.12
1.5.5 Proof of Proposition 1.10
1.5.6 Proof of Proposition 1.13
2 Hypothesis Testing
2.1 Preliminaries from Statistics: Hypotheses, Tests, Risks
2.1.1 Hypothesis Testing Problem
2.1.2 Tests
2.1.3 Testing from repeated observations
2.1.4 Risk of a simple test
2.1.5 Two-point lower risk bound
2.2 Hypothesis Testing via Euclidean Separation
2.2.1 Situation
2.2.2 Pairwise Hypothesis Testing via Euclidean Separation
2.2.3 Euclidean Separation, Repeated Observations, and Majority Tests
2.2.4 From Pairwise to Multiple Hypotheses Testing
2.3 Detectors and Detector-Based Tests
2.3.1 Detectors and their risks
2.3.2 Detector-based tests
2.4 Simple observation schemes
2.4.1 Simple observation schemes-Motivation
2.4.2 Simple observation schemes-The definition
2.4.3 Simple observation schemes-Examples
2.4.4 Simple observation schemes-Main result
2.4.5 Simple observation schemes-Examples of optimal detectors
2.5 Testing multiple hypotheses
2.5.1 Testing unions
2.5.2 Testing multiple hypotheses "up to closeness"
2.5.3 Illustration: Selecting the best among a family of estimates
2.6 Sequential Hypothesis Testing
2.6.1 Motivation: Election polls
2.6.2 Sequential hypothesis testing
2.6.3 Concluding remarks
2.7 Measurement Design in simple observation schemes
2.7.1 Motivation: Opinion polls revisited
2.7.2 Measurement Design: Setup
2.7.3 Formulating the MD problem
2.8 Affine detectors beyond simple observation schemes
2.8.1 Situation
2.8.2 Main result
2.9 Beyond the scope of affine detectors: lifting the observations
2.9.1 Motivation
2.9.2 Quadratic lifting: Gaussian case
2.9.3 Quadratic lifting-Does it help?
2.9.4 Quadratic lifting: Sub-Gaussian case
2.9.5 Generic application: Quadratically constrained hypotheses
2.10 Exercises for Chapter 2
2.10.1 Two-point lower risk bound
2.10.2 Around Euclidean Separation
2.10.3 Hypothesis testing via l1-separation
2.10.4 Miscellaneous exercises
2.11 Proofs
2.11.1 Proof of the observation in Remark 2.8
2.11.2 Proof of Proposition 2.6 in the case of quasi-stationary K-repeated observations
2.11.3 Proof of Theorem 2.23
2.11.4 Proof of Proposition 2.37
2.11.5 Proof of Proposition 2.43
2.11.6 Proof of Proposition 2.46
3 From Hypothesis Testing to Estimating Functionals
3.1 Estimating linear forms on unions of convex sets
3.1.1 The problem
3.1.2 The estimate
3.1.3 Main result
3.1.4 Near-optimality
3.1.5 Illustration
3.2 Estimating N-convex functions on unions of convex sets
3.2.1 Outline
3.2.2 Estimating N-convex functions: Problem setting
3.2.3 Bisection estimate: Construction
3.2.4 Building Bisection estimate
3.2.5 Bisection estimate: Main result
3.2.6 Illustration
3.2.7 Estimating N-convex functions: An alternative
3.3 Estimating linear forms beyond simple observation schemes
3.3.1 Situation and goal
3.3.2 Construction and main results
3.3.3 Estimation from repeated observations
3.3.4 Application: Estimating linear forms of sub-Gaussianity parameters
3.4 Estimating quadratic forms via quadratic lifting
3.4.1 Estimating quadratic forms, Gaussian case
3.4.2 Estimating quadratic form, sub-Gaussian case
3.5 Exercises for Chapter 3
3.6 Proofs
3.6.1 Proof of Proposition 3.3
3.6.2 Verifying 1-convexity of the conditional quantile
3.6.3 Proof of Proposition 3.4
3.6.4 Proof of Proposition 3.14
4 Signal Recovery by Linear Estimation
Overview
4.1 Preliminaries: Executive summary on Conic Programming
4.1.1 Cones
4.1.2 Conic problems and their duals
4.1.3 Schur Complement Lemma
4.2 Near-optimal linear estimation from Gaussian observations
4.2.1 Situation and goal
4.2.2 Building a linear estimate
4.2.3 Byproduct on semidefinite relaxation
4.3 From ellitopes to spectratopes
4.3.1 Spectratopes: Definition and examples
4.3.2 Semidefinite relaxation on spectratopes
4.3.3 Linear estimates beyond ellitopic signal sets and ? ?2-risk
4.4 Linear estimates of stochastic signals
4.4.1 Minimizing Euclidean risk
4.4.2 Minimizing ? ?-risk
4.5 Linear estimation under uncertain-but-bounded noise
4.5.1 Uncertain-but-bounded noise
4.5.2 Mixed noise
4.6 Calculus of ellitopes/spectratopes
4.7 Exercises for Chapter 4
4.7.1 Linear estimates vs. Maximum Likelihood
4.7.2 Measurement Design in Signal Recovery
4.7.3 Around semidefinite relaxation
4.7.4 Around Propositions 4.4 and 4.14
4.7.5 Signal recovery in Discrete and Poisson observation schemes
4.7.6 Numerical lower-bounding minimax risk
4.7.7 Around S-Lemma
4.7.8 Miscellaneous exercises
4.8 Proofs
4.8.1 Preliminaries
4.8.2 Proof of Proposition 4.6
4.8.3 Proof of Proposition 4.8
4.8.4 Proof of Lemma 4.17
4.8.5 Proofs of Propositions 4.5, 4.16 and 4.19
4.8.6 Proofs of Propositions 4.18 and 4.19, and justification of Remark 4.20
5 Signal Recovery Beyond Linear Estimates
Overview
5.1 Polyhedral estimation
5.1.1 Motivation
5.1.2 Generic polyhedral estimate
5.1.3 Specifying sets Hd for basic observation schemes
5.1.4 Efficient upper-bounding of R[H] and contrast design, I
5.1.5 Efficient upper-bounding of R[H] and contrast design, II
5.1.6 Assembling estimates: Contrast aggregation
5.1.7 Numerical illustration
5.1.8 Calculus of compatibility
5.2 Recovering signals from nonlinear observations by Stochastic Optimization
5.2.1 Problem setting
5.2.2 Assumptions
5.2.3 Estimating via Sample Average Approximation
5.2.4 Stochastic Approximation estimate
5.2.5 Numerical illustration
5.2.6 "Single-observation" case
5.3 Exercises for Chapter 5
5.3.1 Estimation by Stochastic Optimization
5.4 Proofs
5.4.1 Proof of (5.8)
5.4.2 Proof of Lemma 5.6
5.4.3 Verification of (5.44)
5.4.4 Proof of Proposition 5.10
Solutions to Selected Exercises
6.1 Solutions for Chapter 1
6.2 Solutions for Chapter 2
6.2.1 Two-point lower risk bound
6.2.2 Around Euclidean Separation
6.2.3 Hypothesis testing via l1 separation
6.2.4 Miscellaneous exercises
6.3 Solutions for Chapter 3
6.4 Solutions for Chapter 4
6.4.1 Linear Estimates vs. Maximum Likelihood
6.4.2 Measurement Design in Signal Recovery
6.4.3 Around semidefinite relaxation
6.4.4 Around Propositions 4.4 and 4.14
6.4.5 Numerical lower-bounding minimax risk
6.4.6 Around S-Lemma
6.4.7 Miscellaneous exercises
6.5 Solutions for Chapter 5
6.5.1 Estimation by Stochastic Optimization
Appendix: Executive Summary on Efficient Solvability of Convex Optimization Problems
Bibliography
Index

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