Mathematical Foundations of Machine Learning, Fall 2020

Instructor
Justin Romberg
Office: Coda S1109
Phone: (404) 894-3930

Description
The purpose of this course is to provide first year PhD students in engineering and computing with a solid mathematical background for two of the pillars of modern data science: linear algebra and applied probability.

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Outline

I. Vector space basics
a) linear vector spaces, linear independence
b) norms and inner products
c) linear approximation
d) basis expansions

II. Regression using least-squares
a) regression as a linear inverse problem
b) the least-squares problem
c) ridge regression
d) regression in a Hilbert space, representer theorem
e) reproducing kernels, kernel regression, Mercer’s theorem

III. Solving and analyzing least-squares problems
a) the Singular Value Decomposition (SVD) and the pseudo­inverse
b) stable inversion and regularization
c) matrix factorization
d) steepest descent and conjugate gradients

IV. Statistical estimation and classification
a) review of core concepts in probability
b) Gaussian estimation
c) maximum likelihood estimation
d) Bayesian estimation
e) the Bayes classifier
e) empirical risk minimization
f) logistic regression

V. Further topics as time permits
a) gradient descent for optimization
b) stochastic gradient descent
c) multi-layer neural networks, the chain rule and back propagation