The Modern Mathematics of Deep Learning

A manifold and a chart

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.

Local structure · Describe charts, local coordinates and localization.

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: 4.1 When data live on a smaller shape

A manifold can have a lower local dimension d′ than its surrounding space ℝᵈ. A chart assigns local coordinates to points on the manifold.

Local coordinates let a function on a manifold be studied in fewer dimensions.

Source formula example

\mathbf{\mathcal{M}}

illustration · The line in ℝ² is one simple manifold example; a general manifold can require several charts.

Symbols

$\mathcal{M}$: A d′-dimensional manifold in an ambient space 𝒳.
$𝒳$: The ambient input space.
$x$: A point on 𝓜.
$d$: Ambient dimension.
$d^{\prime}$: Manifold dimension.
$γᵢ$: A local coordinate chart.

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 = [2, 0]
d = 2
intrinsicDimension = 1

Calculated here

Result expression: { x, d, intrinsicDimension, chartValue: chart(x), belongsToLine: x[1] === 0 }

Returns

Example result

x: array (2 items): [2, 0]
d: 2
intrinsicDimension: 1
chartValue: 2
belongsToLine: 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 line in ℝ² is one simple manifold example; a general manifold can require several charts.

Source: Inline notation