The Modern Mathematics of Deep Learning

Input and output grid groups

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

Represent grid positions by coordinate pairs with wraparound. Downsampling reduces the side length from 2d to d.

These groups specify the positions before and after a convolution layer.

Source formula example

{{G}}=({\mathbb{Z}}/(2d{\mathbb{Z}}))^{2}

illustration · This enumerates positions, not the full convolution or pooling map.

Symbols

$G$: Input group (ℤ/(2dℤ))².
$G̃$: Output group (ℤ/(dℤ))².
$d$: Positive integer output side length.
$ℤ/(kℤ)$: Integer positions modulo k.

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.

Output side d

This example’s data flow

You choose

input.d = 1

Initial values in code

inputPositions = []
outputPositions = []

Calculated here

Result expression: { inputPositions, outputPositions, inputCount: inputPositions.length, outputCount: outputPositions.length, }

d = input.d Uses: input.d
for (let i = 0; i < 2 * d; i++) for (let j = 0; j < 2 * d; j++) inputPositions.push([i, j]); Uses: d, inputPositions
for (let i = 0; i < d; i++) for (let j = 0; j < d; j++) outputPositions.push([i, j]); Uses: d, outputPositions

Returns

Example result

inputPositions: array (4 items): [array (2 items): [0, 0], array (2 items): [0, 1], array (2 items): [1, 0], array (2 items): [1, 1]]
outputPositions: array (1 items): [array (2 items): [0, 0]]
inputCount: 4
outputCount: 1

Some runtime paths or values are not resolved here.

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Coordinates here are zero-based representatives. Addition wraps independently modulo the respective side length.

Source: Source expression