Alphabeta MathAdaptive spectral regularization by SOLIT
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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
- Compact linear inverse maps and the Picard criterion
- Compact Operators and Riesz Schauder Theory
- Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Deterministic source classes and optimal recovery
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite Probability and the Probabilistic Method
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Gaussian sequence inversion and exact risk
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Independence Borel Cantelli and Zero One Laws
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Inverse maps, identifiability, and stability
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Numerical Inverse Problems and Regularization Error
- Order, Zorn's Lemma, and the Axiom of Choice
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Probability Spaces Random Variables and Expectation
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Spectral regularization and parameter choice
- Splitting Fields
- Statistical Experiments, Models and Identifiability
- Statistical inverse experiments and white noise
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Inverse and Implicit Function Theorems
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Weak Laws and Series of Independent Random Variables
- Weak nonparametric Bernstein–von Mises limits
2 · Logical flowchart
3 · Items
Lemma
Gaussian series norm tails and fourth moments
Definition
Ordered spectral filters, source indices, and variance grids
Lemma
Risk decomposition and monotone bias for ordered spectral filters
Algorithm
SOLIT parameter choice on a geometric-variance grid
Theorem
SOLIT oracle risk bound and spectral rates
Sources
- Vladimir Spokoiny and Niklas Willrich, Bootstrap tuning in ordered model selection, Appendix D, Gaussian quadratic forms
- Housen Li and Frank Werner, Adaptive minimax optimality in statistical inverse problems via SOLIT, §§3.1.1–3.2.1
- Christian Clason, Regularization of Inverse Problems, complete lecture notes, Chapter 5 §§5.1–5.2
- Housen Li and Frank Werner, Adaptive minimax optimality in statistical inverse problems via SOLIT, Lemma 3.6 and §§3.1.1–3.2.1
- Christian Clason, Regularization of Inverse Problems, complete lecture notes, Chapter 5 §5.1
- Housen Li and Frank Werner, Adaptive minimax optimality in statistical inverse problems via SOLIT, §§3.2.1–3.2.2
- Vladimir Spokoiny and Niklas Willrich, Bootstrap tuning in ordered model selection, §§2.1–2.4
- Housen Li and Frank Werner, Adaptive minimax optimality in statistical inverse problems via SOLIT, Theorems 3.4, 3.8, 4.1 and 4.3
- Vladimir Spokoiny and Niklas Willrich, Bootstrap tuning in ordered model selection, Theorem 2.3 and Appendix A
- Christian Clason, Regularization of Inverse Problems, complete lecture notes, Chapter 5 §5.2