The Modern Mathematics of Deep Learning

Radial directions and exponent

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

j indexes a direction wⱼ and the corresponding one-variable function ϱⱼ.

A sum of ridge terms uses one function per direction.

Source formula example

j

illustration · This uses ReLU as one possible ridge function; the source permits general functions ϱⱼ.

Symbols

$j$: A one-based direction index.
$wⱼ$: A direction vector.
$ϱⱼ$: The corresponding one-variable ridge function.

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

directions = [ [1, 0], [0, 1], ]
x = [2, 3]
j = 2

Calculated here

Result expression: { directions, x, j, dot, activation: Math.max(0, dot) }

dot = directions[j - 1].reduce((s, w, i) => s + w * x[i], 0) Uses: directions, j, x

Returns

Example result

directions: array (2 items): [array (2 items): [1, 0], array (2 items): [0, 1]]
x: array (2 items): [2, 3]
j: 2
dot: 3
activation: 3

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 ReLU as one possible ridge function; the source permits general functions ϱⱼ.

Source: Inline notation