The Modern Mathematics of Deep Learning

A complex Fourier phase

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.

Functions and derivatives · Describe maps and how their outputs change.

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: 1.2.2 Can the network express the answer?

The complex exponential e^(−2πix) has real part cos(2πx) and imaginary part −sin(2πx).

Fourier transforms use complex phases to encode frequency.

Source formula example

x\mapsto e^{-2\pi ix}

illustration · Floating-point trigonometric calculations can return tiny rounding errors near zero.

Symbols

$x$: A real input.
$i$: The imaginary unit with i² = −1.

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.

This example’s data flow

Initial values in code

x = 0.25

Calculated here

Result expression: { x, real: Math.cos(2 * Math.PI * x), imaginary: -Math.sin(2 * Math.PI * x) }

Returns

Example result

x: 0.25
real: 0
imaginary: -1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. Floating-point trigonometric calculations can return tiny rounding errors near zero.

Source: Inline notation