The Modern Mathematics of Deep Learning

Fourier energy preservation

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

With unitary normalization, the total squared Fourier magnitude equals the total squared input magnitude.

Energy preservation means this representation retains the input’s norm.

Source formula example

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

illustration · This finite identity does not prove Parseval’s theorem for all square-integrable functions.

Symbols

$u$: Square-integrable input signal.
$Ψ$: Representation taking Fourier magnitude.
$\Vert ·\Vert L²$: Square root of integrated squared magnitude.

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 coordinate

This example’s data flow

You choose

input.first = 1

Calculated here

Result expression: { inputNorm, featureNorm }

x = [input.first, 2, 0, 0] Uses: input.first
coefficients = x.map((_, k) => x.reduce( (z, v, j) => [ z[0] + (v * Math.cos((2 * Math.PI * j * k) / 4)) / 2, z[1] - (v * Math.sin((2 * Math.PI * j * k) / 4)) / 2, ], [0, 0], ), ) Uses: x
inputNorm = Math.sqrt(x.reduce((sum, v) => sum + v * v, 0)) Uses: x
featureNorm = Math.sqrt( coefficients.reduce((sum, z) => sum + z[0] ** 2 + z[1] ** 2, 0), ) Uses: coefficients

Returns

Example result

inputNorm: ≈2.236
featureNorm: ≈2.236

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The four-position toy uses the unitary discrete Fourier transform, with factor 1/√4. The source describes the corresponding L² identity.

Source: Source expression