The Modern Mathematics of Deep Learning

Fourier modulus descriptor

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.

Signal transformations · Study signal transforms, scattering and their properties.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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: 7.1 Features that stay steady

The Fourier modulus descriptor Ψ maps a signal u to the absolute value of its Fourier transform.

Fourier magnitudes are unchanged by translations of the signal.

Source formula example

\Psi

illustration · The example uses a finite discrete Fourier transform; it does not establish the continuous-space statement or reconstruction property.

Symbols

$u$: An L² signal.
$Ψ$: The Fourier-modulus descriptor.
$L²(ℝᵈ)$: Square-integrable signals on real d-dimensional space.

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

u = [1, 0, -1, 0]

Calculated here

Result expression: { u, magnitude }

magnitude = u.map((_, k) => { const re = u.reduce((s, v, n) => s + v * Math.cos((2 * Math.PI * k * n) / 4), 0); const im = u.reduce((s, v, n) => s - v * Math.sin((2 * Math.PI * k * n) / 4), 0); return Math.hypot(re, im); }) Uses: u

Returns

Example result

u: array (4 items): [1, 0, -1, 0]
magnitude: array (4 items): [0, 2, 0, 2]

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. The example uses a finite discrete Fourier transform; it does not establish the continuous-space statement or reconstruction property.

Source: Inline notation