Broyden’s Method in Python

In a previous post we looked at root-finding methods for single variable equations. In this post we’ll look at the expansion of Quasi-Newton methods to the multivariable case and look at one of the more widely-used algorithms today: Broyden’s Method.

Continue reading “Broyden’s Method in Python”

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Root-Finding Algorithms Tutorial in Python: Line Search, Bisection, Secant, Newton-Raphson, Inverse Quadratic Interpolation, Brent’s Method

Motivation

How do you find the roots of a continuous polynomial function? Well, if we want to find the roots of something like:

f(x) = x^2 + 3x - 4

Continue reading “Root-Finding Algorithms Tutorial in Python: Line Search, Bisection, Secant, Newton-Raphson, Inverse Quadratic Interpolation, Brent’s Method”

Statistical Learning Theory: VC Dimension, Structural Risk Minimization

Sometimes our models overfit, sometimes they overfit.

A model’s capacity is, informally, its ability to fit a wide variety of functions. As a simple example, a linear regression model with a single parameter has a much lower capacity than a linear regression model with multiple polynomial parameters. Different datasets demand models of different capacity, and each time we apply a model to a dataset we run the risk of overfitting or underfitting our data.

Continue reading “Statistical Learning Theory: VC Dimension, Structural Risk Minimization”

DropConnect Implementation in Python and TensorFlow

I wouldn’t expect DropConnect to appear in TensorFlow, Keras, or Theano since, as far as I know, it’s used pretty rarely and doesn’t seem as well-studied or demonstrably more useful than its cousin, Dropout. However, there don’t seem to be any implementations out there, so I’ll provide a few ways of doing so. Continue reading “DropConnect Implementation in Python and TensorFlow”

Style Transfer with Tensorflow

A Neural Algorithm of Artistic Style” is an accessible and intriguing paper about the distinction and separability of image content and image style using convolutional neural networks (CNNs). In this post we’ll explain the paper and then run a few of our own experiments.

To begin, consider van Gogh’s “The Starry Night”: Continue reading “Style Transfer with Tensorflow”

The Box-Cox Transformation

The Box-Cox transformation is a family of power transform functions that are used to stabilize variance and make a dataset look more like a normal distribution. Lots of useful tools require normal-like data in order to be effective, so by using the Box-Cox transformation on your wonky-looking dataset you can then utilize some of these tools.

Here’s the transformation in its basic form. For value x and parameter \lambda:

\displaystyle \frac{x^{\lambda}-1}{\lambda} \quad \text{if} \quad x\neq 0 

\displaystyle log(x) \quad \text{if} \quad x=0

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