The Modern Mathematics of Deep Learning

Vector and scalar product space

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: 1.2.2 Can the network express the answer? · 1.3 The questions ahead · 4.2 Approximating with random pieces · More source sections

A Cartesian product contains ordered pairs, one value from each named space.

Product spaces describe examples or parameters made of several parts.

Source formula example

({\mathbb{R}}^{d}\times{\mathbb{R}})

illustration · The example has a two-entry vector and one scalar; ℝᵈ × ℝᵏ allows the second part to be a vector instead.

Symbols

$ℝᵈ$: Real d-entry vectors.
$ℝ$: Real scalars.
$x, y$: The two parts of a pair.
$w⁽¹⁾, b⁽¹⁾$: A weight vector and scalar bias.

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

x = [1, 2]
y = 3

Calculated here

Result expression: { pair: [x, y], vectorLength: x.length, scalar: Number.isFinite(y) }

Returns

Example result

pair: array (2 items): [array (2 items): [1, 2], 3]
vectorLength: 2
scalar: 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 example has a two-entry vector and one scalar; ℝᵈ × ℝᵏ allows the second part to be a vector instead.

Source: Inline notation