Cauchy distribution

As explained in Data Without Labels

The Cauchy distribution is a continuous probability distribution with a sharp central peak, slowly decaying heavy tails, and no well-defined mean.

The Cauchy distribution is a continuous probability distribution with a narrow central peak and slowly decaying heavy tails. Unlike the normal distribution, it assigns more probability to values far from the center and has no well-defined mean; its median is the symmetry center. section 6.4.1

In dimensionality reduction, the distribution supports this workflow:

• Calculate low-dimensional similarity probabilities qj|i for counterparts yi and yj of high-dimensional points xi and xj. section 6.4.1
• Compare high-dimensional probabilities pj|i with qj|i using KL divergence. section 6.4 section 6.4.1
• Minimize the divergence with gradient descent so the low-dimensional mapping preserves similarity relationships. section 6.4.1

Figure 6.13 Comparison of Gaussian distribution vs. Cauchy distribution. (Image source: Quora)

Compare the Gaussian and Cauchy curves to see the Cauchy distribution’s narrower peak and heavier tails. section 6.4.1

How Data Without Labels covers it

How deep it goes

Defines shape and central tendency, gives the probability role, and stops short of deriving the distribution’s formula.

Examples

Uses a Gaussian-versus-Cauchy curve comparison and paired high- and low-dimensional points to illustrate similarity probabilities.

What it assumes

Assumes familiarity with probability distributions and introduces the comparison of high- and low-dimensional similarities through KL divergence.

What you can do after

Evaluate how Cauchy-based low-dimensional probabilities support similarity preservation during stochastic neighbor embedding.

How it connects

Connects the Cauchy distribution to normal distribution, stochastic neighbor embedding, KL divergence, and gradient descent.

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Related concepts

• cauchy mutationused by · A mutation method that adds a random value sampled from a Cauchy distribution to a real-valued gene.
• normal distributioncontrasts with · A normal distribution is a continuous probability distribution with a symmetric, bell-shaped density determined by its mean and variance.
• meancontrasts with · A mean is a measure of central tendency, calculated as the arithmetic average of values or, for a random variable, as the probability-weighted average of its possible values.
• t-SNEused by · t-SNE is a nonlinear dimensionality-reduction technique that maps high-dimensional data to a lower-dimensional space while emphasizing local similarities or neighborhood relationships.
• inverse cumulative distribution function methodimplemented by · The inverse cumulative distribution function method generates samples from a target univariate distribution by applying its inverse cumulative distribution function to uniformly distributed random values.

This explainer was generated by Manning's knowledge graph from the text of Data Without Labels. The section links point to the parts of the book it draws on.