The Modern Mathematics of Deep Learning

Deforming an input

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

At position x, the deformed signal uses the original signal at x minus the displacement τ(x).

A feature representation can be studied for sensitivity to these changes.

Source formula example

u\mapsto u_{\tau}:=u(\cdot-\tau(\cdot))

illustration · This pointwise example does not establish deformation stability.

Symbols

$u$: Input function.
$uτ$: Deformed function.
$τ$: Position-dependent displacement function.
$x$: Input location.

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.

Displacement rate

This example’s data flow

You choose

input.rate = 0.25

Initial values in code

x = 2

Calculated here

Result expression: { x, displacement, original, deformed }

displacement = input.rate * x Uses: input.rate, x
original = Math.exp(-x * x) Uses: x
deformed = Math.exp(-((x - displacement) ** 2)) Uses: x, displacement

Returns

Example result

x: 2
displacement: 0.5
original: ≈0.01832
deformed: ≈0.1054

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The example uses the square-integrable Gaussian u(x) = exp(−x²) and τ(x) = rate times x. The source discusses smooth vector-valued deformations.

Source: Source expression