The Modern Mathematics of Deep Learning

Diagonal encoder coefficients

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.

Frames and sparse signals · Encode or reconstruct signals with frames and dictionaries.

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: 6.3 Taking apart and rebuilding

When singular vectors are paired with a diagonal matrix, cross-coefficients vanish.

These operations define structured network layers and their analysis.

Source formula example

C_{i,j}=\langle x,u_{i}\ast_{\circ}v_{j}\rangle=\langle u_{i},\mathbb{H}_{n}(x)v_{j}\rangle=0,\quad i\neq j.

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

Symbols

$Cᵢⱼ$: Analysis coefficient for indices i and j.
$uᵢ,vⱼ$: Paired left and right singular vectors of the window matrix.
$Hₙ(x)$: Window matrix of x.

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 singular value

This example’s data flow

You choose

input.s = 1

Calculated here

Result expression: { coefficients: H, offDiagonal: [H[0][1], H[1][0]] }

H = [ [input.s, 0], [0, 2], ] Uses: input.s

Returns

Example result

coefficients: array (2 items): [array (2 items): [1, 0], array (2 items): [0, 2]]
offDiagonal: array (2 items): [0, 0]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This diagonal example takes both sets of singular vectors as the standard basis. The source identifies singular-vector pairs of the window matrix; arbitrary frames need not give zero cross-coefficients.

Source: Source expression