MTH 5810 - Mathematical Methods for Machine Learning

4 lecture hours 0 lab hours 4 credits
Course Description
This course surveys the essential linear algebra and multivariate calculus required for graduate study in machine learning. Topics include matrix algebra, real vector spaces, inner product spaces, differentiation and Newton's method, partial differentiation, the gradient, the chain rule, and optimization of multivariate functions.
Prereq: Enrollment in machine learning graduate program
Note: This course is not open to undergraduate students.
Course Learning Outcomes
Upon successful completion of this course, the student will be able to:
  • Describe linear and affine subsets of three-dimensional space both algebraically and geometrically
  • Interpret the value of a dot product geometrically
  • Interpret functions of two variables via the corresponding surface, level curves, and contour plot
  • Perform elementary operations with matrices including manipulating expressions and equations involving matrices
  • Determine if a matrix is invertible and find its inverse in this case
  • Solve systems of linear equations using matrix methods and technology
  • Express the solutions of a consistent matrix equation in parametric vector form
  • Determine if a subset of n-dimensional space is a subspace
  • Find the matrix of a linear transformation and bases for its kernel and range
  • Show that a real number is an eigenvalue of a matrix with real number entries by finding the corresponding eigenvectors
  • Determine the eigenvalues of a matrix using technology
  • Use the inner product to find the length of a vector and the distance between two vectors
  • Determine if a set of vectors is an orthogonal basis for a subspace of R^n
  • Find the projection of a vector onto a subspace of R^n
  • Find a general linear model by solving the normal equation
  • Find and interpret the singular value decomposition of a matrix using technology
  • Use numerical methods to locate extrema of a single-variable function
  • Perform operations with series representations of the single-variable elementary functions
  • Find first and higher-order partial derivatives of a function
  • Interpret partial derivatives as rates of change in applications
  • Find the total differential of a function of more than one variable, use it to estimate change
  • Estimate error propagation using the total differential
  • Construct the matrix form of the multivariate chain rule and use it to find a derivative
  • Find the gradient of a function and interpret its direction
  • Find directional derivatives of a function and interpret the results
  • Determine the maximum, minimum, and saddle points on a surface
  • Interpret mathematics from machine learning literature

Prerequisites by Topic
  • Single-variable calculus

Course Topics
  • Geometry in three dimensions: lines, planes, distance, surfaces, vectors, and the dot product
  • Matrix algebra
  • Matrix equations and invertibility
  • Real vector spaces
  • Linear transformations and their matrices
  • Inner products, orthogonality, projection, and applications
  • Eigenvectors and eigenvalues and interpretation of matrix factorizations
  • Review of single-variable differentiation
  • Taylor series of elementary functions
  • Finding critical points analytically and numerically
  • Partial derivatives
  • Gradients, directional derivatives, and the Jacobian
  • Extrema of functions of two variables

Coordinator
Dr. Anthony van Groningen


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