The Modern Mathematics of Deep Learning

Number sets

Penguin learning project

The project uses 342 complete Palmer penguin records: 240 training-pool and 102 held-out rows. The selected training rows determine feature scaling and model parameters. Source CSV

This project trains weights with softmax, unlike the homepage lab, which uses nearest species averages. Same penguin measurements; different classification methods. Predictions can differ.

Sets and notation · Describe domains, indices and membership.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

Consider this concept alongside the project; its source assumptions may differ. These observations do not prove a theorem.

Input: 12 × 2; Held-out evaluation: 102 · #58 · 40.6 mm, 193 mm. Known answer: Adélie; Prediction: Adélie · [≈0.7383, ≈0.2444, ≈0.01737].

Used in these source sections: 1.1 A note on symbols

ℕ denotes natural numbers, ℤ denotes integers including negative values and zero, and ℝ denotes real numbers.

In this book, indices and counts use positive integers.

Sets state which values are allowed in a formula.

Source formula example

\mathbb N,\quad\mathbb Z,\quad\mathbb R

exact · JavaScript uses finite-precision numbers. Number.isFinite checks a representable finite value, not arbitrary mathematical real numbers.

Source formula example

This source example uses its own inputs and assumptions. Editing it does not change the project.

Edit a value or the code, then run the example. This is a finite demonstration, not a proof of the paper’s theorem.

Number

This example’s data flow

You choose

input.value = 2

Calculated here

Result expression: { integer: Number.isInteger(n), positiveInteger: Number.isInteger(n) && n >= 1, finiteReal: Number.isFinite(n), }

n = input.value Uses: input.value

Returns

Example result

integer: true
positiveInteger: true
finiteReal: true

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The symbol ℕ is used for positive counts in this book. Other authors may include zero. A mathematical set ℝ also includes values that cannot be represented exactly by JavaScript numbers.

Source: Teaching example