The Modern Mathematics of Deep Learning

A radial target 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

Here the radial function g depends on a vector through its squared Euclidean length. The scalar function g̃ takes that nonnegative squared length as input.

This structure reduces a vector input to one scalar.

Source formula example

\tilde{g}\colon[0,\infty)\to{\mathbb{R}}

illustration · This uses g̃(r)=r²; general radial functions need not have that form.

Symbols

$g$: The radial function on ℝᵈ.
$g̃$: A function from nonnegative squared radii to real outputs.
$d$: Input dimension.

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

x = [3, 4]

Calculated here

Result expression: { x, squaredRadius, value: gTilde(squaredRadius) }

squaredRadius = x.reduce((s, v) => s + v * v, 0) Uses: x

Returns

Example result

x: array (2 items): [3, 4]
squaredRadius: 25
value: 625

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 uses g̃(r)=r²; general radial functions need not have that form.

Source: Inline notation