The Modern Mathematics of Deep Learning

Unit ball in Euclidean space

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.

Sets and notation · Describe domains, indices and membership.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 4.2 Approximating with random pieces · 6.5 Removing connections

The open unit ball around zero contains vectors with Euclidean length less than one.

The ball restricts the domain to vectors near the origin.

Source formula example

B_{1}(0)\subset{\mathbb{R}}^{d}

illustration · The calculation checks one point in two dimensions.

Symbols

$x$: A real vector.
$B₁(0)$: The open Euclidean ball with center zero and radius one.
$d$: The positive integer 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 = [0.3, 0.4]

Calculated here

Result expression: { x, norm, belongs: norm < 1 }

norm = Math.sqrt(x.reduce((s, v) => s + v * v, 0)) Uses: x

Returns

Example result

x: array (2 items): [0.3, 0.4]
norm: 0.5
belongs: true

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 calculation checks one point in two dimensions.

Source: Inline notation