The Modern Mathematics of Deep Learning

Values indexed by a set

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.

Vectors and matrices · Combine numbers using vector and matrix operations.

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 · 6.3 Taking apart and rebuilding

ℝᴳ means a real value is assigned to every index in G. A list can represent these assignments when G is finite and ordered.

Indexed arrays represent signals and filters.

Source formula example

a,b\in{\mathbb{R}}^{{G}}

illustration · The example uses three ordered indices; the source can use other finite cyclic index sets.

Symbols

$G, H$: Finite index sets.
$x, uᵢ, vⱼ, a, b$: Real-valued indexed arrays on the indicated set.
$ℝᴳ$: All real-valued functions on G.

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

G = [0, 1, 2]
x = [2, 4, 6]

Calculated here

Result expression: { G, x, pairs: G.map((g, i) => [g, x[i]]) }

Returns

Example result

G: array (3 items): [0, 1, 2]
x: array (3 items): [2, 4, 6]
pairs: array (3 items): [array (2 items): [0, 2], array (2 items): [1, 4], array (2 items): [2, 6]]

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 example uses three ordered indices; the source can use other finite cyclic index sets.

Source: Inline notation