Saturday, October 29, 2016

Unsupervised Learning II: the power of improper convex relaxations

In this continuation post I'd like to bring out a critical component of learning theory that is essentially missing in today's approaches for unsupervised learning: improper learning by convex relaxations.

Consider the task of learning a sparse linear separator. The sparsity, or $\ell_0$, constraint is non-convex and computationally hard. Here comes the Lasso - replace the $\ell_0$ constraint with $\ell_1$ convex relaxation --- et voila --- you've enabled polynomial-time solvable convex optimization.

Thursday, October 6, 2016

A mathematical definition of unsupervised learning?

Extracting structure from data is considered by many to be the frontier of machine learning. Yet even defining "structure", or the very goal of learning structure from unlabelled data, is not well understood or rigorously defined.

In this post we'll give a little background to unsupervised learning and argue that, compared to supervised learning, we lack a well-founded theory to tackle even the most basic questions. In the next post, we'll introduce a new candidate framework.