The Modern Mathematics of Deep Learning

Convolution and modulus cascade

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

Apply a circular convolution and take absolute values, then repeat and average.

These computations organize features and sparse coefficients.

Source formula example

|||u\ast\psi_{j_{1}}|\ast\psi_{j_{2}}\ast\dots|\ast\psi_{j_{\ell}}|\ast\varphi_{J},

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$u$: Input signal.
$ψⱼ$: Wavelet filter at index j.
$φⱼ$: Low-pass averaging filter.
$∗$: Convolution.
$|·|$: Pointwise modulus.
$ℓ$: Number of cascade stages.
$j$: Index of a wavelet filter.
$J$: Index of the low-pass scale.

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.

First value

This example’s data flow

You choose

input.first = 1

Initial values in code

waveletIndex = 0
lowpassScale = 0

Calculated here

Result expression: { waveletIndex, lowpassScale, first, second, pooled: conv(second, [0.5, 0.5]) }

x = [input.first, 2] Uses: input.first
first = conv(x, [1, -1]).map(Math.abs) Uses: conv, x
second = conv(first, [1, -1]).map(Math.abs) Uses: conv, first

Returns

Example result

first: array (2 items): [1, 1]
second: array (2 items): [0, 0]
pooled: array (2 items): [0, 0]
waveletIndex: 0
lowpassScale: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This invented two-position circular cascade illustrates repeated convolution, modulus, and averaging. It does not establish stability, invariance, or norm preservation for the scattering transform. The toy labels its selected wavelet j = 0 and low-pass scale J = 0; these labels do not imply a particular source filter family.

Source: Source expression