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
  • Neural Networks by Grant Sanderson (3Blue1Brown)
    video series available online
  • Geometry of Deep Learning by Jong Chul Ye
    available via Dartmouth Libraries (link)
  • All of Statistics: A Concise Course on Statistical Inference by Larry Wasserman
    available online
  • Essence of Linear Algebra by Grant Sanderson (3Blue1Brown)
    video series available online
  • 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: Knowledge and reason

Keywords: tables, functions, frames, semantic networks, knowledge graphs

Reading material:

  • Neapolitan et al. Chapter 4 "Certain Knowledge Representation" (book chapter)


Wed Jun 26

Lecture: Probabilistic graphical models

Keywords: directed acyclic graphs, Bayesian networks, Markov random fields (MRFs), factor graphs

Reading material:

  • Neapolitan et al. Chapter 7 "Uncertain Knowledge Representation" (book chapter)


Thu Jun 27

Lecture: Inference on probabilistic graphical models

Keywords: conditional probabilities, random variables, Bayes' theorem, message passing / belief propagation

Reading material:

  • Wasserman Chapter I.1 "Probability" and Chapter I.2 "Random Variables" (book)


Fri Jun 28

Lecture: Linear regression

Keywords: linear regression, train-test split, quality of fit, residual sum of squares (RSS), mean squared error (MSE), R squared, ordinary least squares (OLS), Gauss-Markov theorem, collinearity, heteroscedasticity

Reading material:


Mon Jul 1

Lecture: Review of linear algebra

Keywords:vectors, matrices, linear transformations, change of basis, eigenbasis, eigendecomposition, singular value decomposition

Reading material:

  • Sanderson (3Blue1Brown) Chapter 3 "Linear Transformations" (video)
  • Sanderson (3Blue1Brown) Chapter 7 "Inverse matrices, column space and null space" (video)
  • Sanderson (3Blue1Brown) Chapter 13 "Change of basis" (video)
  • Sanderson (3Blue1Brown) Chapter 14 "Eigenvectors and eigenvalues" (video)
  • Brunton et al. Section 1.3 "Singular Value Decomposition: Mathematical Overview" (video, book)


Wed Jul 3

Lecture: Regression and classification

Keywords: the bias-variance tradeoff, 1-hot encoding, discriminative models, logistic regression, k-nearest neighbors (KNN), generative models, decision boundary, linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), naive Bayes

Reading material:


Thu Jul 4

No class on Independence Day.




Fri Jul 5

Lecture: Resampling and validation

Keywords: holdout set, validation accuracy, crossvalidation, leave-one-out crossvalidation (LOOCV), bootstrap, data leakage, imputation

Reading material:


Mon Jul 8

Lecture: Feature selection

Keywords: shrinkage, best subset selection, stepwise forward selection, stepwise backward selection

Reading material:


Wed Jul 10

Lecture: Regularization

Keywords: curse of dimensionality, Lp norm, gradient descent, ridge regression, lasso, elastic net, feature engineering, dimension reduction, principal component analysis (PCA), principal component regression (PCR)

Reading material:


Thu Jul 11

Code lab




Fri Jul 12

Lecture: Decision trees

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

Reading material:


Mon Jul 15

Lecture: Support vector machines

Keywords: linearly separable patterns, maximum-margin models, hard margin, soft margin, support vector classifier, loss functions for binary classification

Reading material:


Wed Jul 17

Lecture: Function approximation

Keywords:feature mapping, feature augmentations, linear classifiers with non-linear decision boundaries, manifold learning, infinite-dimensional vector spaces, function spaces, random features, Koopman operator

Reading material:


Thu Jul 18

Code lab




Fri Jul 19

Lecture: Kernel methods

Keywords: linear kernels, non-linear kernels, kernel trick, Hilbert spaces, reproducing-kernel Hilbert spaces, representer theorems

Reading material:


Mon Jul 22

Lecture: Introduction to neural networks: Perceptron and beyond

Keywords: perceptron, activation function, ReLU, perceptron learning algorithm, stochastic gradient descent, epoch, mini-batch, momentum, RMSProp, Adam

Reading material:


Wed Jul 24

Lecture: Neural network architectures and neural coding

Keywords: feed-forward neural network, softmax, deep learning, encoder, decoder, convolutional neural networks (CNNs)

Reading material:

o


Thu Jul 25

Code lab




Fri Jul 26

Lecture: Training and regularizing neural networks

Keywords: backpropagation, double descent, drop out, early stopping, regularization

Reading material:


Mon Jul 29

Lecture: Forecasting and prediction

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

Reading material:


Wed Jul 31

Lecture: Natural language processing

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

Reading material:

  • TBD


Thu Aug 1

Code lab




Fri Aug 2

Lecture: Natural language processing (continued)

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

Reading material:

  • TBD


Mon Aug 5

Project proposals




Wed Aug 7

Lecture: Representation learning

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

Reading material:


Thu Aug 8

Code lab




Fri Aug 9

Lecture: Clustering

Keywords: k-means clustering, hierarchical clustering

Reading material:


Mon Aug 12

Lecture: The topology of data

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

Reading material:

  • TBD


Wed Aug 14

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)


Thu Aug 15

Code lab




Fri Aug 16

Lecture: Network analysis

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

Reading material:

  • TBD


Mon Aug 19

Final project presentations




Wed Aug 21

Final project presentations




Lecture: Transfer learning

Keywords: teacher-student learning, multitask learning

Reading material:

  • TBD

Lecture: Image generation and more transfer learning

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

Reading material:

  • TBD

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.