Bounded Rationality (Posts about manifolds)http://bjlkeng.github.io/enSat, 03 Aug 2024 01:42:51 GMTNikola (getnikola.com)http://blogs.law.harvard.edu/tech/rssHyperbolic Geometry and Poincaré Embeddingshttp://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/Brian Keng<div><p>This post is finally going to get back to some ML related topics.
In fact, the original reason I took that whole math-y detour in the previous
posts was to more deeply understand this topic. It turns out trying to
under tensor calculus and differential geometry (even to a basic level) takes a
while! Who knew? In any case, we're getting back to our regularly scheduled program.</p>
<p>In this post, I'm going to explain one of the applications of an abstract
area of mathematics called hyperbolic geometry. The reason why this area is of
interest is because there has been a surge of research showing its
application in various fields, chief among them is a paper by Facebook
researchers [1] in which they discuss how to utilize a model of hyperbolic
geometry to represent hierarchical relationships. I'll cover some of
the math weighting more towards intuition, show some of their results, and also
show some sample code from Gensim. Don't worry, this time I'll try much harder
not going to go down the rabbit hole of trying to explain all the math (no
promises though).</p>
<p>(Note: If you're unfamiliar with tensors or manifolds, I suggest getting a quick
overview with my previous two posts:
<a class="reference external" href="http://bjlkeng.github.io/posts/tensors-tensors-tensors/">Tensors, Tensors, Tensors</a> and
<a class="reference external" href="http://bjlkeng.github.io/posts/manifolds/">Manifolds: A Gentle Introduction</a>)</p>
<p><a href="http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/">Read more…</a> (34 min remaining to read)</p></div>embeddingsgeometryhyperbolicmanifoldsmathjaxPoincaréhttp://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/Sun, 17 Jun 2018 12:20:18 GMTManifolds: A Gentle Introductionhttp://bjlkeng.github.io/posts/manifolds/Brian Keng<div><p>Following up on the math-y stuff from my <a class="reference external" href="http://bjlkeng.github.io/posts/tensors-tensors-tensors/">last post</a>,
I'm going to be taking a look at another concept that pops up in ML: manifolds.
It is most well-known in ML for its use in the
<a class="reference external" href="https://www.quora.com/What-is-the-Manifold-Hypothesis-in-Deep-Learning">manifold hypothesis</a>.
Manifolds belong to the branches of mathematics of topology and differential
geometry. I'll be focusing more on the study of manifolds from the latter
category, which fortunately is a bit less abstract, more well behaved, and more
intuitive than the former. As usual, I'll go through some intuition,
definitions, and examples to help clarify the ideas without going into too much
depth or formalities. I hope you mani-like it!</p>
<p><a href="http://bjlkeng.github.io/posts/manifolds/">Read more…</a> (30 min remaining to read)</p></div>manifoldsmathjaxmetric tensorhttp://bjlkeng.github.io/posts/manifolds/Tue, 17 Apr 2018 11:24:57 GMT