Instructor: Dr. Alice Schwarze (alice.c.schwarze@dartmouth.edu)

Classes: (2) MWF 2:10 - 3:15 and x-hour Th 1:20 - 2:10 in Kemeny 007

Office hour: W 3:15 - 4:15 in Kemeny 342

Invitation to the Dartmouth Workshop in 1956; source: https://raysolomonoff.com/dartmouth/

Mathematics and AI offers an exploration of the intersection between mathematics and artificial intelligence (AI). Covering state-of-the-art machine learning techniques and their mathematical foundations, this course aims to provide students with both a broad theoretical understanding and practical skills. The syllabus starts with a brief review of the history of AI, and current limits and issues. This is followed by an introduction to statistical learning in a supervised setting and a deeper dive on neural networks and their applications with some references to current mathematical research. The syllabus continues with an overview of unsupervised learning methods and their applications in feature selection. It concludes with student's presentations of their final projects.

Prerequisite courses and skills: Math 13, Math 22 or Math 24, and Math 23, or advanced placement/ instructor override. Familiarity with at least one programming language. (Python preferred.) Students who request an instructor override should have encountered the concepts in the Prerequisite concepts checklist in their previous coursework or self study.

Prerequisite concepts: derivative of a function, chain rule, smooth function, optimization, Taylor expansion, differential equation, fixed point, vector, matrix, lines, curves, subspaces, eigenvector, eigenvalue multivariate function, partial derivative, spherical coordinates, probability distribution, conditional probability, joint probability

Textbooks

  • Introduction to Statistical Learning with Applications in Python by Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani, Jonathan Taylor
    available on the book's website
  • Data-Driven Science and Engineering by Steven Brunton and Nathan Kutz
    videos and PDF (download) available on the book's website
  • Geometry of Deep Learning by Jong Chul Ye
    available via Dartmouth Libraries (link)
  • Artificial Intelligence With an Introduction to Machine Learning by Richard Neapolitan and Xia Jiang
    available via Dartmouth Libraries (link)
  • Limits of AI - Theoretical, Practical, Ethical by Klaus Mainzer and Reinhard Kahle
    available via Dartmouth Libraries (link)

Other materials

  • J. Veisdal: "The Birthplace of AI" (2024) (substack post)
  • A. Turing: "Computing Machinery and Intelligence" (1950) (full paper)

The following is a tentative schedule for the course. Please check back regularly for updates as the term progresses.

Thu Jun 20

No class




Fri Jun 21

Lecture: Artificial intelligence: Ideas and their evolution

Keywords: Turing test, Dartmouth workshop, expert systems, strong AI, weak AI, artificial general intelligence (AGI)

Reading material:

  • Neapolitan et al. Chapter 1.1 "History of Artificial Intelligence" (book chapter)
  • J. Veisdal: "The Birthplace of AI" (2024) (substack post)
  • A. Turing: "Computing Machinery and Intelligence" (1950) §1 "The Imitation Game" (full paper)


Mon Jun 24

Lecture: Representing knowledge

Keywords: tables, functions, frames, knowledge graphs, causal networks, directed acyclic graphs, Bayesian networks, Markov random fields (MRFs)

Reading material:

  • Neapolitan et al. Chapter 4 "Certain Knowledge Representation" (book chapter)
  • Neapolitan et al. Chapter 7 "Uncertain Knowledge Representation" (book chapter)
  • Additional reading to review probability: Neapolitan et al. Chapter 6.1 "Probability Basics" and Chapter 6.2 "Random Variables" (book chapter)


Wed Jun 26

Lecture: Linear regression

Keywords: linear regression, gradient descent, mean squared error

Reading material:


Thu Jun 27

Lecture: Regression and classification

Keywords: logistic regression, k-nearest neighbors (KNN), linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), naive Bayes, 1-hot encoding

Reading material:


Fri Jun 28

Lecture: Resampling and validation

Keywords: Crossvalidation, bootstrap, data leakage

Reading material:


Mon Jul 1

Lecture: Feature selection

Keywords: subset selection, shrinkage, dimension reduction, principal component regression (PCR)

Reading material:


Wed Jul 3

Lecture: Regularization

Keywords: ridge regression, lasso

Reading material:


Thu Jul 4

No class on Independence Day.




Fri Jul 5

Lecture: Basis functions for regression

Keywords: step functions, splines, radial basis functions (RBFs), generalized additive models (GAMs)

Reading material:


Mon Jul 8

Lecture: Decision trees

Keywords: regression trees, classification trees, tree ensemble methods, bagging, boosting, random forests, Bayesian additive regression trees (BART)

Reading material:


Wed Jul 10

Lecture: Support vector machines

Keywords: maximum-margin models, hard margin, soft margin, VC theory, nonlinear kernels

Reading material:


Thu Jul 11

Lecture: Kernel methods

Keywords: kernel trick, kernel ridge regression, reproducing kernel Hilbert spaces, representer theorems

Reading material:

  • TBD


Fri Jul 12

Catch-up and Review




Mon Jul 15

Lecture: Introduction to neural networks: Perceptron and beyond

Keywords: perceptron, multi-class perceptron, universal approximation theorems, ReLU, softmax

Reading material:


Wed Jul 17

Lecture: Neural network architectures and neural coding

Keywords: feed-forward neural network, deep learning, encoder, decoder

Reading material:


Thu Jul 18

Lecture: Training and regularizing neural networks

Keywords: backpropagation, stochastic gradient descent, Adam, drop out

Reading material:


Fri Jul 19

Lecture: Transfer learning

Keywords: teacher-student learning, multitask learning

Reading material:

  • TBD


Mon Jul 22

Lecture: Forecasting and prediction

Keywords: Taken's theorem, time-delayed embedding, recurrent neural networks (RNNs), reservoir computing

Reading material:


Wed Jul 24

Lecture: Natural language processing

Keywords: structured prediction, text classification, bag of words, self-supervised learning, word embeddings

Reading material:

  • TBD


Thu Jul 25

Lecture: Natural language processing (continued)

Keywords: long-term short-term memory, attention, transformer, generative pre-trained transformers (GPTs)

Reading material:

  • TBD


Fri Jul 26

Lecture: Image generation and more transfer learning

Keywords: general adversarial networks (GANs), contrastive language-image pre-training (CLIP), DALL-E, diffusion

Reading material:

  • TBD


Mon Jul 29

Project proposals




Wed Jul 31

Lecture: Representation learning

Keywords: latent space, autoencoders, restricted Boltzmann machines (RBMs)

Reading material:


Thu Aug 1

No class




Fri Aug 2

Lecture: Principal component analysis

Keywords: principal component analysis (PCA), matrix factorizations, Hebbian learning

Reading material:


Mon Aug 5

Project updates




Wed Aug 7

Lecture: The topology of data

Keywords: self-organizing maps (SOMs), competitive learning, topological data analysis (TDA)

Reading material:

  • TBD


Thu Aug 8

No class




Fri Aug 9

Lecture: Clustering

Keywords: k-means clustering, hierarchical clustering

Reading material:


Mon Aug 12

Project updates




Wed Aug 14

Lecture: Network analysis

Keywords: centrality measures, community detection, modularity maximization, belief propagation

Reading material:

  • TBD


Thu Aug 15

No class




Fri Aug 16

Lecture: Matrix completion

Keywords: Low rank matrix completion, high rank matrix completion, link prediction, recommender systems

Reading material:

  • Brunton et al. Section 1.5 (video)


Mon Aug 19

Final project presentations




Wed Aug 21

Final project presentations




In person lectures and office hours

Lectures and office hours will generally be held in person. From time to time, it may be announced in class or on CANVAS that lectures or office hours will be conducted via zoom. Individual appointments with your instructor may be held remotely via zoom, especially those made for late afternoon.

Grading

The course grade will be based upon on

  • weekly in-class quizzes (total 50 points),
  • weekly homework problems (total 100 points),
  • class participation, which may involve presenting a blog post or research paper in class, (total 100 points),
  • and a final coding project to be completed by the final week of classes (total 250 points). This project can be a group project.

Academic Honor Principle

For quizzes and all other assessments, Dartmouth's Academic Honor Principle will be upheld. Please be advised of especially

  • Quizzes: Giving and/or receiving assistance during an examination violates the Academic Honor Principle.
  • Homework and course projects: Collaboration is both permitted and encouraged, but it is a violation of the honor code for someone to provide the answers for you.

Student Religious Observances

Some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with your instructor before the end of the second week of the term to discuss appropriate accommodations.