The Modern Mathematics of Deep Learning

Scattering feature norm

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 the energy of an input with the energy of a representation.

Norm preservation means feature magnitudes retain information about input magnitude.

Source formula example

\|\Psi(u)\|\approx\|u\|_{L^{2}}

illustration · One finite, truncated scattering example does not establish a norm preservation theorem.

Symbols

$u$: Input signal with finite L² norm.
$Ψ(u)$: Scattering representation.
$\Vert ·\Vert$: Norm of the representation.
$≈$: Approximate relation, not exact equality.

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 input value

This example’s data flow

You choose

input.value = 1

Initial values in code

average = [0.5, 0.5]
wavelet = [1, -1]

Calculated here

Result expression: { lowpass, waveletFeatures, inputNorm: norm(u), featureNorm: norm(features) }

u = [input.value, input.value] Uses: input.value
lowpass = conv(u, average) Uses: conv, u, average
modulus = conv(u, wavelet).map(Math.abs) Uses: conv, u, wavelet
waveletFeatures = conv(modulus, average) Uses: conv, modulus, average
features = [...lowpass, ...waveletFeatures] Uses: lowpass, waveletFeatures

Returns

Example result

lowpass: array (2 items): [1, 1]
waveletFeatures: array (2 items): [0, 0]
inputNorm: ≈1.414
featureNorm: ≈1.414

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented two-position circular signal is constant, and the finite representation contains its low-pass output and one wavelet-modulus-averaged output. Their norms coincide in this case. The source discusses the full scattering transform under appropriate assumptions.

Source: Source expression