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

Eckart-Young: the Best Low-Rank Approximation

Chapter 8 · Mathematics for AI and Machine Learning · Advanced · 1:43

An interactive live film: the columns of a 3 x 40 matrix are a cloud; a rank-1 matrix projects them onto a line, a rank-2 onto a plane, with the residuals as red segments. Turn the line until the error is within 2% of the Eckart-Young minimum.

Narration

Here is a matrix A with three rows and forty columns. Each column is a point in space, so the matrix is this cloud.

A matrix of rank one has every column on one line through the origin. The closest such matrix to A projects every point onto the line, and its error is the sum of the squared distances: the red segments.

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

More from Chapter 8

Interactive1:49SVD: Rotate, Stretch, RotateChapter 8Preview1:31Energy from Singular Values: Rebuild a Picture Layer by LayerChapter 8Preview1:33Double Descent at the Interpolation ThresholdChapter 8Preview1:32The Marchenko-Pastur Law and the Escaping EigenvalueChapter 8Preview1:46The Pseudoinverse: the Shortest SolutionChapter 8Preview1:31Polar Decomposition: Stretch, then RotateChapter 8Preview1:47Randomized SVD: Catching the Top DirectionsChapter 8