The Modern Mathematics of Deep Learning

Distance and balls

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.

Norms and distances · Measure size or difference in a specified space.

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: 1.1 A note on symbols

A distance assigns a nonnegative number to each pair of points. An open ball contains points whose distance from x is less than r.

A ball describes points near a center.

Source formula example

({\mathcal{X}},d_{\mathcal{X}})

illustration · This invented finite example illustrates the notation; it does not establish a general theorem.

Symbols

$𝒳$: A set of points.
$d𝒳$: A distance on 𝒳.
$x$: The center, in 𝒳.
$r$: A positive radius.

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.

Value

This example’s data flow

You choose

input.x = 2

Initial values in code

points = [0, 1, 2, 3]
radius = 2

Calculated here

Result expression: { center, radius, ball: points.filter((y) => Math.abs(y - center) < radius) }

center = input.x Uses: input.x

Returns

Example result

center: 2
radius: 2
ball: array (3 items): [1, 2, 3]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values in this example are invented. This invented finite example illustrates the notation; it does not establish a general theorem.

Source: Inline notation