This article follows the steps of Henri Lebesgue as he came upon his theory of integration. The story could be started earlier, but we don't lose too much by starting with Borel, Lebesgue's adviser, at the end of the 19th century. Borel and the measure of a set At the end of the 19th century, Émile Borel was thinking about the problem of measure, that is, the problem of describing the size of things. Read more...

A lot of machine learning techniques can be viewed as an attempt to represent high-dimensional data in fewer dimensions without losing any important information. In a sense, it is lossy compression—compressing the data to be small and amenable before being passed to some next stage of data processing. If our data consists of elements of \( \mathbb{R}^D \), we are trying to find interesting functions of the form: \[ f : \mathbb{R}^D \to \mathbb{R}^d \] where \( D \) is fixed, but \( d \) can be chosen freely. Read more...

Whenever I come across a matrix multiplication, my first attempt at visualizing it is to view the multiplication as: multiple objects, combined together, many times Matrices are usually carrying a list of objects, with each object represented by a row or column of the matrix. Inspecting how matrices behave by looking these objects can be an effective way to understand what an author is trying to communicate when they use matrices. Read more...

Reading matrix notation is burdened by the trivial things like rows being indexed before columns. An author can be trying to communicate something simple, yet the reader's cognitive load can be high as they unpack the matrix notation. Here, I'm experimenting with ways to make matrix notation more memorable. Indexing There is no fundamental reason why rows should appear before columns in matrix indexing. It's just a convention to be remembered. Read more...