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Machine Learning the Hard Way

Smooth manifolds II: differentiability and smoothness

This is the second part of an introductory post on differentiable manifolds. The previous post restricted attention to a class of spaces – the topological manifolds – which locally look like \mathbb{R}^n. We saw that the existence of local homeomorphisms to \mathbb{R}^n is sufficient to represent a space locally using real coordinates. Similarly,…

Smooth manifolds I: Topological manifolds

Some motivation from machine learning This post introduced the concept of smoothness for functions of the form f:\mathbb{R}^n\rightarrow\mathbb{R}. This relied on viewing both \mathbb{R}^n and \mathbb{R} as smooth manifolds – places where we could do calculus. In doing things ‘the hard way’, it becomes unavoidable to talk about general smooth manifolds. While…

Linear transformations and bases

This post will cover: Previous posts covered special classes of functions between real spaces \mathbb{R}^n and \mathbb{R}^m, namely continuous functions, and the subclasses of continuous functions which are additionally differentiable or smooth. Defining continuity required viewing real spaces with their metric topology, while defining smoothness additionally required viewing them as differentiable manifolds, places…

Smooth scalar fields and RSS loss

The previous post focused on the structure of \mathbb{R}^n as a topological space given its metric topology. The topological structure allows us to define continuous functions, and hence we could define continuous scalar fields. We demonstrated that in the two examples of scalar fields from this post, the logistic classifier was special in that…


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About Me

I’m Zach. I am currently starting my career in Data Science at NYC Data Science Academy. Before that, I got a PhD in linguistics from UMD, MsED in math eduction at St. John’s, and BFA in film from NYU Tisch. I have worked as a teacher, recording engineer, and researcher.

This blog focuses on concepts in Machine Learning and Data Science from a more traditional mathematical viewpoint. Read more here.

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