The Modern Mathematics of Deep Learning

Multiple positive dimensions

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: 1.2.1 Finding better settings · 1.2.2 Can the network express the answer? · 3.2 Simple pieces, many layers · More source sections

The symbols before ∈ are each positive integers. They can represent different dimensions or orders in the surrounding statement.

Positive integers specify counts and sizes.

Source formula example

d,n\in{\mathbb{N}}

illustration · The finite numbers are examples; they do not establish the surrounding theorem.

Symbols

$d, n, k$: Positive integers; their roles are fixed by the surrounding statement.
$ℕ$: The natural numbers.

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.

This example’s data flow

Initial values in code

d = 2
n = 3

Calculated here

Result expression: { d, n, valid: [d, n].every((v) => Number.isInteger(v) && v > 0) }

Returns

Example result

d: 2
n: 3
valid: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. The finite numbers are examples; they do not establish the surrounding theorem.

Source: Inline notation