The Modern Mathematics of Deep Learning

An index map between grids

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

A map can send every input position to an output-grid position.

This example makes the source notation or condition concrete.

Source formula example

p:{{G}}\to\widetilde{{{G}}}

illustration · The source does not specify this position map as the only possible downsampling rule.

Symbols

$G$: Input position group.
$G̃$: Output position group.
$p$: A map between these position sets.
$d$: Positive output side length.

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.

Zero-based input row

This example’s data flow

You choose

input.row = 2

Initial values in code

d = 2

Calculated here

Result expression: { inputSide: 2 * d, outputSide: d, inputPosition, outputPosition }

inputPosition = [input.row, 3] Uses: input.row
outputPosition = inputPosition.map((v) => Math.floor(v / 2)) Uses: inputPosition

Returns

Example result

inputSide: 4
outputSide: 2
inputPosition: array (2 items): [2, 3]
outputPosition: array (2 items): [1, 1]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source states p:G→G̃ and gives a separate downsampling operation on grid-valued data. The invented position map here groups pairs of coordinates using floor(i/2); it illustrates the declared domain and codomain, without claiming this is the unique source choice.

Source: Source expression