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
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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
Github repository: https://github.com/acuschwarze/mathematics-and-ai/
Canvas: https://canvas.dartmouth.edu/courses/66608/
Webpage: https://math.dartmouth.edu/~m76x24/
The following is a tentative schedule for the course. Please check back regularly for updates as the term progresses.
Keywords: Turing test, Dartmouth workshop, expert systems, strong AI, weak AI, artificial general intelligence (AGI)
Reading material:
Keywords: tables, functions, frames, knowledge graphs, causal networks, directed acyclic graphs, Bayesian networks, Markov random fields (MRFs)
Reading material:
Keywords: linear regression, gradient descent, mean squared error
Reading material:
Keywords: logistic regression, k-nearest neighbors (KNN), linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), naive Bayes, 1-hot encoding
Reading material:
Keywords: Crossvalidation, bootstrap, data leakage
Reading material:
Keywords: subset selection, shrinkage, dimension reduction, principal component regression (PCR)
Reading material:
Keywords: ridge regression, lasso
Reading material:
Keywords: step functions, splines, radial basis functions (RBFs), generalized additive models (GAMs)
Reading material:
Keywords: regression trees, classification trees, tree ensemble methods, bagging, boosting, random forests, Bayesian additive regression trees (BART)
Reading material:
Keywords: maximum-margin models, hard margin, soft margin, VC theory, nonlinear kernels
Reading material:
Keywords: kernel trick, kernel ridge regression, reproducing kernel Hilbert spaces, representer theorems
Reading material:
Keywords: perceptron, multi-class perceptron, universal approximation theorems, ReLU, softmax
Reading material:
Keywords: feed-forward neural network, deep learning, encoder, decoder
Reading material:
Keywords: backpropagation, stochastic gradient descent, Adam, drop out
Reading material:
Keywords: teacher-student learning, multitask learning
Reading material:
Keywords: Taken's theorem, time-delayed embedding, recurrent neural networks (RNNs), reservoir computing
Reading material:
Keywords: structured prediction, text classification, bag of words, self-supervised learning, word embeddings
Reading material:
Keywords: long-term short-term memory, attention, transformer, generative pre-trained transformers (GPTs)
Reading material:
Keywords: general adversarial networks (GANs), contrastive language-image pre-training (CLIP), DALL-E, diffusion
Reading material:
Keywords: latent space, autoencoders, restricted Boltzmann machines (RBMs)
Reading material:
Keywords: principal component analysis (PCA), matrix factorizations, Hebbian learning
Reading material:
Keywords: self-organizing maps (SOMs), competitive learning, topological data analysis (TDA)
Reading material:
Keywords: k-means clustering, hierarchical clustering
Reading material:
Keywords: centrality measures, community detection, modularity maximization, belief propagation
Reading material:
Keywords: Low rank matrix completion, high rank matrix completion, link prediction, recommender systems
Reading material:
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.
The course grade will be based upon on
For quizzes and all other assessments, Dartmouth's Academic Honor Principle will be upheld. Please be advised of especially
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.