Convolutions are a mathematical tool of translation invariant transformations to objects. It’s best to think about 1 dimensional convolutions before thinking about convolutions of 2 and 3 dimensions. A convolution of two functions \(x[n]\) and \(h[n]\) produces a new function \(y[n] = (x\star h)[n]\). The operation \(y[n] = \sum_{k=0}^{N-1}x[k]h[n-k]\) where the \(x\) has support over […]
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Dilated Convolutions
In this post, I wanted to try a new type of problem. This is the problem of semantic segmentation. Semantic segmentation is essentially a method of assigning categories to different parts of an image. The best way to explain it is through a picture: There are several methods to find accurate segmentations of an image. There […]
Evaluation of LEM and T-SNE
From the last two posts, we saw some pretty pictures from the Laplacian Eigenmaps (LEM) when we applied it to the Kepler dataset. We looked primarily at avalanche distributions in the last two posts, but we never actually plotted any distributions! I wanted to remedy that by showing the temperature plot (which showed the greatest […]
The Geometry behind Manifold Learning
In this short blog post, I wanted to review the geometry behind the manifold learning techniques from the previous post. Recalling from the previous post, the generic manifold learning algorithm follows 2 steps: Assign the data to a graph based on the similarity between samples. Here, the similarity might mean the Euclidean metric, or it […]
Manifold Learning
In this blog post, we will apply the methods of nonlinear dimensionality reduction (also known as manifold learning) to the Kepler dataset, using the paradigm of avalanches as described in the first post. The entire set of code is on github, please check it out! We have about \(16,000\) stars to process, with each star containing a […]
Avalanches and Astrophysics
As a first post, I want to go over some data paradigms that I will be using later other blog posts. The Kepler spacecraft (as opposed to satellite, as I learned) orbits the sun looking for exoplanets on other stars. It does this using the transit method of exoplanet observation. The transit method is a simple way […]