The Modern Mathematics of Deep Learning

A tangent 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.

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

The tangent space Tₓ𝓜 contains the directions tangent to the manifold at x.

Tangent spaces describe local directions along a manifold.

Source formula example

T_{x}\mathcal{M}

illustration · For this straight line in ℝ², tangent vectors have zero second coordinate; general manifolds have position-dependent tangent spaces.

Symbols

$\mathcal{M}$: The manifold.
$x$: A point on it.
$Tₓ\mathcal{M}$: Its tangent space at x.

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

point = [2, 0]
coefficient = 3

Calculated here

Result expression: { point, tangent, tangentDimension: 1 }

tangent = [coefficient, 0] Uses: coefficient

Returns

Example result

point: array (2 items): [2, 0]
tangent: array (2 items): [3, 0]
tangentDimension: 1

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. For this straight line in ℝ², tangent vectors have zero second coordinate; general manifolds have position-dependent tangent spaces.

Source: Inline notation