The interactive film is in the book's Companion

Play it, pause it, turn it and drag it with Mathematics for AI and Machine Learning.

Get the book

Block Cholesky: Two Steps to a Circle

Chapter 10 · Mathematics for AI and Machine Learning · Advanced · 1:40

An interactive live film: block Cholesky factors a covariance matrix into triangular blocks; inverting it whitens the distribution in two steps: scaling the first block and shearing out the correlation with the Schur complement, morphing a tilted ellipse into a round circle.

Narration

A block matrix can be factored just like a regular matrix. For a symmetric positive definite matrix, block Cholesky factors it into lower triangular blocks times their transpose.

Look at this tilted covariance ellipse over x zero and x one. Its tilt comes from the off diagonal block, the correlation between the two variables.

The rest of the narration is in the book's Companion.

More from Chapter 10

Interactive2:11Tensor Parallelism: Block Products Across DevicesChapter 10Preview1:24Block Diagonal Matrices: Separate WorldsChapter 10Preview1:28Block Multiplication: Blocks Multiply Like NumbersChapter 10Preview1:39Longer Contexts with Rotary PositionsChapter 10Preview2:20Woodbury Identity: a Rank-One Update of the InverseChapter 10Preview1:40Schur Complement: What Is Left After Minimizing Out a BlockChapter 10Preview1:32Kronecker Product and the Vectorization IdentityChapter 10Preview1:30Online Softmax, One Tile at a TimeChapter 10Preview1:56Mixture of Experts: Routing Tokens to a Few BlocksChapter 10