The Modern Mathematics of Deep Learning

A zero-deformation comparison

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

Compare feature outputs before and after a zero displacement.

This calculation explains the corresponding source expression.

Source formula example

\|\Psi(u)-\Psi(u_{\tau})\|\lesssim K(\tau,\nabla\tau,\nabla^{2}\tau)

illustration · Zero displacement checks a trivial finite case and cannot establish the source deformation stability claim.

Symbols

$Ψ$: Scattering feature descriptor.
$uτ$: Deformed input.
$τ$: Smooth displacement.
$∇τ,∇²τ$: First and second displacement derivatives.
$K$: Unspecified source function of the displacement and its derivatives.
$≲$: Comparison up to an unspecified constant.

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.

Constant signal value

This example’s data flow

You choose

input.value = 1

Initial values in code

displacement = [0, 0]

Calculated here

Result expression: { displacement, featureBefore, featureAfter, difference, toyK: 0 }

signal = [input.value, input.value] Uses: input.value
lowpass = signal.map(() => (signal[0] + signal[1]) / 2) Uses: signal
featureBefore = [...lowpass, 0, 0] Uses: lowpass
featureAfter = [...lowpass, 0, 0] Uses: lowpass
difference = Math.sqrt( featureBefore.reduce((s, v, i) => s + (v - featureAfter[i]) ** 2, 0), ) Uses: featureBefore, featureAfter

Returns

Example result

displacement: array (2 items): [0, 0]
featureBefore: array (4 items): [1, 1, 0, 0]
featureAfter: array (4 items): [1, 1, 0, 0]
difference: 0
toyK: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy uses a constant two-position circular signal, zero displacement, and the finite low-pass plus wavelet-modulus features from the scattering example; the difference is exactly zero. For zero deformation we choose a toy K value 0. The source does not specify an explicit K formula or numerical comparison constant, so this is not their evaluation.

Source: Source expression