Linear Regression

Great article! Reblogging it for future reference.

Math ∩ Programming

Machine learning is broadly split into two camps, statistical learning and non-statistical learning. The latter we’ve started to get a good picture of on this blog; we approached Perceptrons, decision trees, and neural networks from a non-statistical perspective. And generally “statistical” learning is just that, a perspective. Data is phrased in terms of independent and dependent variables, and statistical techniques are leveraged against the data. In this post we’ll focus on the simplest example of this, linear regression, and in the sequel see it applied to various learning problems.

As usual, all of the code presented in this post is available on this blog’s Github page.

The Linear Model, in Two Variables

And so given a data set we start by splitting it into independent variables and dependent variables. For this section, we’ll focus on the case of two variables, $latex X, Y$. Here, if we want…

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The Fast Fourier Transform

Math ∩ Programming

It’s often said that the Age of Information began on August 17, 1964 with the publication of Cooley and Tukey’s paper, “An Algorithm for the Machine Calculation of Complex Fourier Series.” They published a landmark algorithm which has since been called the Fast Fourier Transform algorithm, and has spawned countless variations. Specifically, it improved the best known computational bound on the discrete Fourier transform from $latex O(n^2)$ to $latex O(n \log n)$, which is the difference between uselessness and panacea.

Indeed, their work was revolutionary because so much of our current daily lives depends on efficient signal processing. Digital audio and video, graphics, mobile phones, radar and sonar, satellite transmissions, weather forecasting, economics and medicine all use the Fast Fourier Transform algorithm in a crucial way. (Not to mention that electronic circuits wouldn’t exist without Fourier analysis in general.) Before the Fast Fourier Transform algorithm was public knowledge, it simply…

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l0-Norm, l1-Norm, l2-Norm, … , l-infinity Norm

Helpful for beginners

Rorasa's blog

I’m working on things related to norm a lot lately and it is time to talk about it. In this post we are going to discuss about a whole family of norm.

What is a norm?

Mathematically a norm is a total size or length of all vectors in a vector space  or matrices. For simplicity, we can say that the higher the norm is, the bigger the (value in) matrix or vector is. Norm may come in many forms and many names, including these popular name: Euclidean distance, Mean-squared Error, etc.

Most of the time you will see the norm appears in a equation like this:

$latex \left \| x \right \|$ where $latex x $ can be a vector or a matrix.

For example, a Euclidean norm of a vector $latex

a = \begin{bmatrix}
\end{bmatrix}$ is $latex \left \| a \right…

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