The Modern Mathematics of Deep Learning

Fourier transform and finite samples

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: 1.1 A note on symbols · 3.1 A rule based on distance

The hat denotes the Fourier transform of g in this source passage. A discrete Fourier transform gives frequency coefficients for a finite list.

Frequency coefficients describe how a signal varies.

Source formula example

\hat{g}

illustration · This finite discrete transform illustrates frequency coefficients; it is not the Fourier transform of a general tempered distribution.

Symbols

$g$: A tempered distribution in the source.
$\hat{g}$: Its Fourier transform.

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

samples = [1, 0, -1, 0]

Calculated here

Result expression: { samples, coefficients }

coefficients = samples.map((_, k) => ({ real: samples.reduce((s, x, n) => s + x * Math.cos((2 * Math.PI * k * n) / 4), 0), imaginary: samples.reduce( (s, x, n) => s - x * Math.sin((2 * Math.PI * k * n) / 4), 0, ), })) Uses: samples

Returns

Example result

samples: array (4 items): [1, 0, -1, 0]
coefficients: array (4 items): [object (2 items), object (2 items), object (2 items), object (2 items)]

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. This finite discrete transform illustrates frequency coefficients; it is not the Fourier transform of a general tempered distribution.

Source: Inline notation