The Modern Mathematics of Deep Learning

Deformations and derivatives

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.

Signal transformations · Study signal transforms, scattering and their properties.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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: 7.1 Features that stay steady

A deformation τ maps positions to displacement vectors. Its first and second derivatives describe how the displacement varies.

These derivatives appear in a bound on changes to a feature descriptor.

Source formula example

\tau\colon{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}

illustration · This is a scalar one-dimensional example; the general source deformation and its derivatives can be vector- and tensor-valued.

Symbols

$τ$: A smooth displacement map.
$∇τ$: Its first derivative or Jacobian.
$∇²τ$: Its second derivative tensor.

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
secondDerivative = 0.2

Calculated here

Result expression: { x, tau, firstDerivative, secondDerivative }

tau = 0.1 * x * x Uses: x
firstDerivative = 0.2 * x Uses: x

Returns

Example result

x: 2
tau: 0.4
firstDerivative: 0.4
secondDerivative: 0.2

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. This is a scalar one-dimensional example; the general source deformation and its derivatives can be vector- and tensor-valued.

Source: Inline notation