The Modern Mathematics of Deep Learning

Fourier transform of a ridge function

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.

Approximation and depth · Study approximation error and network size under stated conditions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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

A ridge function depends on one projection of its input. In the source’s continuous Fourier expression, its transform is supported along the corresponding frequency direction.

The toy uses a 2-by-2 discrete signal constant in its second coordinate. Its transformed values vanish off the first frequency axis.

This structure underlies the source argument about shallow-network approximation.

Source formula example

\hat{\Phi}=\sum_{j=1}^{n}(\hat{\varrho}_{j}\otimes\delta_{{\mathbb{R}}^{d-1}})\circ R_{w_{j}},

illustration · The toy uses a finite normalized transform. The source formula uses continuous Fourier transforms and Dirac distributions; the example does not prove that distributional identity.

Symbols

$\hatΦ$: Fourier transform of the network function.
$δ$: A Dirac distribution in the coordinates orthogonal to the ridge direction.
$R_{w_j}$: A rotation aligning the ridge direction with the first axis.

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.

Signal amplitude

This example’s data flow

You choose

input.amplitude = 1

Calculated here

Result expression: { signal, transform }

signal = [ [input.amplitude, input.amplitude], [0, 0], ] Uses: input.amplitude
transform = [0, 1].map((k1) => [0, 1].map((k2) => { let sum = 0; for (let i = 0; i < 2; i++) for (let j = 0; j < 2; j++) sum += signal[i][j] * (-1) ** (k1 * i + k2 * j); return sum / 2; }), ) Uses: signal

Returns

Example result

signal: array (2 items): [array (2 items): [1, 1], array (2 items): [0, 0]]
transform: array (2 items): [array (2 items): [1, 0], array (2 items): [1, 0]]

Some runtime paths or values are not resolved here.

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source represents the network as a sum of functions depending on one-dimensional projections, then uses Fourier linearity, rotations and tensor products. The toy uses the first axis directly; its other frequency coordinate has only the zero-frequency component.

Source: Ridge Fourier transform