The Modern Mathematics of Deep Learning

Fourier transforms of products

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: 3.1 A rule based on distance

The source studies the Fourier transform of products Φφ and gφ, where φ is the square root of a density.

Frequency representations help analyze approximation error.

Source formula example

\widehat{\Phi\varphi}

illustration · The example shows one coefficient of a discrete transform; it does not establish a continuous Fourier support statement.

Symbols

$Φ,g$: A network function and target function.
$φ$: Square root of the density.
$hat$: 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

f = [1, 2, 3, 4]
phi = [0.5, 0.5, 0.5, 0.5]

Calculated here

Result expression: { f, phi, product, realAtZero }

product = f.map((v, i) => v * phi[i]) Uses: f, phi
realAtZero = product.reduce((a, b) => a + b, 0) Uses: product

Returns

Example result

f: array (4 items): [1, 2, 3, 4]
phi: array (4 items): [0.5, 0.5, 0.5, 0.5]
product: array (4 items): [0.5, 1, 1.5, 2]
realAtZero: 5

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 shows one coefficient of a discrete transform; it does not establish a continuous Fourier support statement.

Source: Inline notation