Warm-up

Consider the equations $\alpha x^2 + \beta x = \gamma$ and the fixed point iteration $\beta x_{k+1} = \gamma - \alpha x_k^2$.

Write an error iteration. Can you give conditions on the coefficients and the initial error that are sufficient to guarantee convergence?

Logistics

• Reaction paper - March 15
  • May do in small groups if you want (up to 3)
  • Problem spec and some paper pointers online
  • SIAM CSE has lots of relevant talks (and often you can chase down pointers)
• Project proposal - March 22
  • Again in small groups (1-3)
  • Not required to be the same direction as reaction paper
  • Use the discussion board to advertise!

Rand NLA comments

• Sketching vs preconditioning

Matrix data

• Matrix data and (approximate) low-rank structure
  • Example data: images, docs, graphs, preferences, joint distributions
  • Interpretation: latent factors and bilinear maps
• Supervised learning applications
  • Imputation and missing data (e.g. collaborative filtering)
  • Guided dimension reduction (e.g. linear discriminant analysis)
• Unsupervised learning applications
  • k-means clustering as a matrix factorization (and alternating iteration)
  • Latent-space embeddings (e.g. LSI)
  • Latent Dirichlet Allocation, NMF, and interepretable topic models
• Example: Linear Discriminant Analysis
  • Basic setup
  • Key numerical issues (robustness and performance)

Activity (submit a notebook or PDF for points)

2021-03-02 notebook