The Modern Mathematics of Deep Learning

Balls and volume in one dimension

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: 2.2 Weight size and room to decide · 3.1 A rule based on distance

A ball contains points within a chosen distance of its center. In one dimension, the open ball of radius r around zero is the interval (−r,r).

The toy chooses r=1/2, so its length is 1. It also checks membership in the unit ball.

The paper uses unit balls as input domains and a volume-one ball in a Fourier construction.

Source formula example

B_r(0)\subset\mathbb R^d,\quad\operatorname{vol}(B_r(0))=1

illustration · The code uses one-dimensional open balls. The volume formula 2r does not apply in higher dimensions; boundary conventions can affect membership but not the interval’s length.

Symbols

$r$: A positive radius.
$d$: Positive dimension.
$B_r(0)$: Ball centered at zero with radius r.

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.

x

This example’s data flow

You choose

input.x = 0.25

Initial values in code

radius = 0.5

Calculated here

Result expression: { x, radius, insideVolumeOneBall: Math.abs(x) < radius, volume: 2 * radius, insideUnitBall: Math.abs(x) < 1, }

x = input.x Uses: input.x

Returns

Example result

x: 0.25
radius: 0.5
insideVolumeOneBall: true
volume: 1
insideUnitBall: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Balls and volume