The Modern Mathematics of Deep Learning

Cyclic index representatives

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.

Sets and notation · Describe domains, indices and membership.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 6.1 Looking locally

One-based indices can represent the same cyclic positions as residues modulo d.

This example explains the corresponding source notation.

Source formula example

[d]\cong{\mathbb{Z}}/(d{\mathbb{Z}})

illustration · The enumeration illustrates the identification for one finite d.

Symbols

$[d]$: One-based positions 1 through d.
$ℤ/(dℤ)$: Integers modulo d.
$[d]²$: Pairs of one-based positions.
$≅$: Structure-preserving identification.

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.

Grid side

This example’s data flow

You choose

input.d = 2

Calculated here

Result expression: { d, oneBased, residues, grid }

d = input.d Uses: input.d
oneBased = Array.from({ length: d }, (_, i) => i + 1) Uses: d
residues = oneBased.map((i) => i % d) Uses: oneBased, d
grid = oneBased.flatMap((i) => oneBased.map((j) => [i % d, j % d])) Uses: oneBased, d

Returns

Example result

d: 2
oneBased: array (2 items): [1, 2]
residues: array (2 items): [1, 0]
grid: array (4 items): [array (2 items): [1, 1], array (2 items): [1, 0], array (2 items): [0, 1], 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

The position d represents residue 0. Pairing representatives gives the two-dimensional cyclic grid; group addition wraps modulo d.

Source: Source expression