# Mathematics for Machine Learning

Eberhard Karls University of Tübingen via YouTube

## Syllabus

Welcome!.

(A) Linear algebra 1: Vector spaces.

(A) Linear algebra 2: Basis and dimension.

(A) Linear algebra 3: Direct sum.

(A) Linear algebra 4: Linear maps, kernel, range.

(A) Linear algebra 5: Matrices.

(A) Linear algebra 6: Invertible maps and matrices.

(A) Linear algebra 7: Transpose.

(A) Linear algebra 8: Change of basis.

(A) Linear algebra 9: Rank of a matrix.

(A) Linear algebra 10: Quotient space; equivalence relation.

(A) Linear algebra 11: Determinant.

(A) Linear algebra 12: Eigenvalues.

(A) Linear algebra 13: Characteristic polynomial.

(A) Linear algebra 14: Trace of a matrix.

(A) Linear algebra 15: Diagonalization; triangular matrices.

(A) Linear algebra 16: Metric spaces.

(A) Linear algebra 17: Normed spaces; p-norms.

(A) Linear algebra 18: Norms on R^d are equivalent.

(A) Linear algebra 19: Convex set induces a norm.

(A) Linear algebra 20: Spaces of continuous and differentiable functions.

(A) Linear algebra 21: Lp-spaces of integrable functions.

(A) Linear algebra 22: Scalar product.

(A) Linear algebra 23: Orthogonal vectors and basis.

(A) Linear algebra 24: Orthogonal matrices.

(A) Linear algebra 25: Symmetric matrices.

(A) Linear algebra 26: Spectral theorem for symmetric matrices.

(A) Linear algebra 27: Positive definite matrices.

(A) Linear algebra 28: Variational characterization of eigenvalues.

(A) Linear Algebra 29: Singular value decomposition.

(A) Linear Algebra 30: Rank-k-approximation, matrix norms.

(A) Linear Algebra 31: Pseudo-inverse of a matrix.

(A) Linear Algebra 32: Continuous linear functionals and operator norm.

(A) Linear Algebra 33: Dual space, Riesz representation theorem.

(C) Calculus 1: Sequences and convergence.

(C) Calculus 2: Continuity.

(C) Calculus 3: Sequences of functions; pointwise and uniform convergence.

(C) Calculus 4: Differentiation on R.

(C) Calculus 5: Riemann integral on R.

(C) Calculus 6: Fundamental theorem of calculus on R.

(C) Calculus 7: Power series.

(C) Calculus 8: Taylor series.

(C) Calculus 9: Sigma-Algebra.

(C) Calculus 10: Measure.

(C) Calculus 11: Lebesgue measure on R^n.

(C) Calculus 12: A set that is not Lebesgue-measurable.

(C) Calculus 13: The Lebesgue integral on R^n.

(C) Calculus 14: Differentiation on R^n: partial derivatives.

(C) Calculus 15: Differentiation on R^n: total derivative.

(C) Calculus 16: Differentiation on R^n: directional derivative.

(C) Calculus 17: Differentiation on R^n: Higher order derivatives.

(C) Calculus 18: Minima, maxima, saddlepoints.

(C) Calculus 19: Matrix calculus.

(P) Probability theory 1: Definition of a probability measure.

(P) Probability theory 2: Different types of measures: discrete, with density; Radon-Nikodym.

(P) Probability Theory 3: Different types of measures: singular measures, Lebesgue decomposition.

(P) Probability Theory 4: Cumulative distribution function.

(P) Probability Theory 5: Random variables.

(P) Probability Theory 6: Conditional Probabilities.

(P) Probability Theory 7: Bayes theorem.

(P) Probability Theory 8: Independence.

(P) Probability Theory 9: Expectation (discrete case).

(P) Probability Theory 10: Variance, covariance, correlation (discrete case).

(P) Probability Theory 11: Expectation and covariance (general case).

(P) Probability Theory 12: Markov and Chebyshev inequality.

(P) Probability theory 13: Example distributions: binomial, poisson, multivariate normal.

(P) Probability theory 14: Convergence of random variables.

(P) Probability theory 15: Borel-Cantelli.

(P) Probability theory 16: Law of large numbers, Central limit theorem.

(P) Probability theory 17: Concentration inequalities.

(P) Probability theory 18: Product space and joint distribution.

(P) Probability theory 19: Marginal distribution.

(P) Probability theory 20: Conditional distribution.

(P) Probability theory 21: Conditional expectation.

(S) Statistics 1: Estimation, bias, variance.

(S) Statistics 2: Confidence sets.

(S) Statistics 3: Maximum likelihood estimator.

(S) Statistics 4: Sufficiency, identifiability.

(S) Statistics 5: Hypothesis testing, level and power of a test.

(S) Statistics 6: Likelihood-ratio tests and Neyman-Pearson lemma.

(S) Statistics 7: p-values.

(S) Statistics 8: Multiple testing.

(P) Probability 17a: Glivenko-Cantelli theorem.

(S) Statistics 9: Non-parametric tests (rank and permutation tests, Kolmogorov-Smirnov).

(S) Statistics 10: The bootstrap.

(S) Statistics 11: Bayesian statistics.

(H) High-dimensional spaces.

Statistical Machine Learning Part 7a - What is a convex optimization problem?.

Statistical Machine Learning Part 15 - Convex optimization, Lagrangian, dual problem.

### Taught by

Tübingen Machine Learning