Alphabeta MathMinimax signal detection and separation rates
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Central Limit Theorems
- Characteristic Functions Inversion and Continuity
- Compact linear inverse maps and the Picard criterion
- Compact Operators and Riesz Schauder Theory
- Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
- Compactness
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- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
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- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
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- The Lebesgue and Riemann Integrals Compared
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- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Urysohn's Lemma and the Tietze Extension Theorem
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Convergence Tightness and Representation
- Weak Laws and Series of Independent Random Variables
- Weak nonparametric Bernstein–von Mises limits
2 · Logical flowchart
3 · Items
Lemma
Weighted chi-square triangular-array limit
Lemma
Sparse Gaussian scalar certificate and extremal profile
Lemma
A finite-array normal limit from vanishing fourth moments
Definition
Minimax detection risk and separation rate
Lemma
Existence of a finite q=2 ellipsoid extremal profile for small radius
Lemma
The q=2 ellipsoid extremal profile minimizes the fourth-power objective
Lemma
Symmetric Gaussian product-mixture testing bound
Theorem
Sparse q<2 detection regime split
Theorem
q=2 Gaussian-sequence ellipsoid detection boundary
Corollary
Sharp Gaussian asymptotics under a vanishing largest weight
Sources
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, equations (4.2)–(4.3)
- Evarist Giné and Richard Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.3
- Yuri I. Ingster and Irina A. Suslina, Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies, ESAIM Probability and Statistics 4 (2000), pp. 53–135, especially pp. 87–100
- Rick Durrett, Probability: Theory and Examples, Version 5 (11 January 2019), section 3.4.2
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §2, PDF pp. 4–6
- Evarist Giné and Richard Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.1 and §6.2.3
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, Remark 4.1
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, equations (4.4)–(4.8)
- Clément Marteau and Theofanis Sapatinas, A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection, §3.1
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.1, PDF pp. 6–7
- Evarist Giné and Richard Nickl, Mathematical Foundations of Infinite-Dimensional Statistical Models, §6.2.3, PDF pp. 506–508
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §4.7, Theorem 4.7, PDF pp. 18–19
- Yuri I. Ingster and Irina A. Suslina, Nonparametric Goodness-of-Fit Testing Under Gaussian Models, Chapters 4 and 6, bibliographic context only; not used as an unread proof premise
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, §2 and §4.1, PDF pp. 4–7
- Clément Marteau and Theofanis Sapatinas, A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection, §§3.1–3.2
- Yuri I. Ingster, Theofanis Sapatinas and Irina A. Suslina, Minimax signal detection in ill-posed inverse problems, Theorem 4.1 and the discussion after equation (4.3), PDF pp. 6–7