The Modern Mathematics of Deep Learning

A density from Fourier energy

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

The source constructs a probability density by choosing a function φ whose Fourier transform is an indicator of a ball of volume one. Energy preservation gives total mass one.

The toy uses a two-point normalized discrete Fourier transform and indicator [1,0].

The construction supplies the probability measure in the depth-separation argument.

Source formula example

\begin{split}\mu({\mathbb{R}}^{d})=\|\varphi^{2}\|_{L^{1}({\mathbb{R}}^{d})}=\|\varphi\|^{2}_{L^{2}({\mathbb{R}}^{d})}=\|\hat{\varphi}\|^{2}_{L^{2}({\mathbb{R}}^{d})}=\|\mathds{1}_{B_{r}(0)}\|^{2}_{L^{2}({\mathbb{R}}^{d})}=1.\end{split}

illustration · This is a finite discrete transform, not the continuous indicator of a Euclidean ball or a proof of Plancherel’s theorem.

Symbols

$φ$: A function whose squared magnitude supplies a density in the source construction.
$μ$: The resulting probability measure.

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

fourier = [1, 0]

Calculated here

Result expression: { fourier, phi, density, mass: density.reduce((a, b) => a + b, 0), fourierEnergy: fourier.reduce((sum, value) => sum + value * value, 0), }

scale = Math.sqrt(2)
phi = [(fourier[0] + fourier[1]) / scale, (fourier[0] - fourier[1]) / scale] Uses: fourier, scale
density = phi.map((value) => value * value) Uses: phi

Returns

Example result

fourier: array (2 items): [1, 0]
phi: array (2 items): [≈0.7071, ≈0.7071]
density: array (2 items): [0.5, 0.5]
mass: 1
fourierEnergy: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source chooses a ball of volume one and uses its indicator as the Fourier transform of φ. The continuous norm identity relies on Plancherel. The finite toy instead uses a unit-energy indicator in a two-point transform.

Source: Density construction