Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Prerequisites
6.046J / 18.410J Design and Analysis of Algorithms or equivalent and 6.041SC Probabilistic Systems Analysis and Applied Probability or 18.440 Probability and Random Variables or equivalent. You will need a strong background in algorithms, probability and linear algebra.
Description
Modern machine learning systems are often built on top of algorithms that do not have provable guarantees, and it is the subject of debate when and why they work. In this class, we will focus on designing algorithms whose performance we can rigorously analyze for fundamental machine learning problems. We will cover topics such as: Nonnegative matrix factorization, tensor decomposition, sparse coding, learning mixture models, matrix completion and inference in graphical models. Almost all of these problems are computationally hard in the worstcase and so developing an algorithmic theory is about (1) choosing the right models in which to study these problems and (2) developing the appropriate mathematical tools (often from probability, geometry or algebra) in order to rigorously analyze existing heuristics, or to design fundamentally new algorithms.
Textbook
There is no textbook for this course. Instead, lecture notes and readings are provided.
Course Outline

Nonnegative Matrix Factorization
 Qualitative Comparisons to SVD
 New Algorithms via Separability
 Applications to Topic Models

Tensor Decompositions
 Tensor Rank, Border Rank and the Rotation Problem
 Jennrich’s Algorithm and the Generalized Eigenvalue Problem
 Learning HMMs
 Mixed Membership Models and Community Detection
 Cumulants and Independent Component Analysis

Sparse Coding
 Sparse Recovery, Incoherence and Uncertainty Principles
 Alternating Minimization via Approximate Gradient Descent
 SumofSquares and Noisy Tensor Decomposition

Learning Mixture Models
 Expectation Maximization
 Clustering in Highdimensions
 Method of Moments and Systems of Polynomial Equations

Linear Inverse Problems
 Nuclear Norm, Atomic Norm and Matrix Completion
 Alternating Minimization via Principal Angles
 Tensor Prediction and Random CSPs
Assessment
Students will be expected to solve 2 problem sets, and complete a researchoriented final project. This could be either a survey, or original research; students will be encouraged to find connections between the course material and their own research interests.