The Modern Mathematics of Deep Learning

Multiple signal channels

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.

Features · Transform inputs into features used by a rule.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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 · 7.2 A few building blocks at a time

A multi-channel signal has one real array on G for each of c channels. With d positions this is dc real values when flattened.

Channel and position dimensions determine the input shape.

Source formula example

({\mathbb{R}}^{G})^{C}

illustration · This example has two channels and three positions; the source can use different counts.

Symbols

$x$: A multi-channel signal.
$G$: A finite position set of size d.
$c, C$: Positive integer channel counts.
$ℝᵈᶜ$: Real vectors with dc entries.

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

channels = [ [1, 2, 3], [4, 5, 6], ]

Calculated here

Result expression: { channels, c, d, flat: channels.flat(), dimension: c * d }

c = channels.length Uses: channels
d = channels[0].length Uses: channels

Returns

Example result

channels: array (2 items): [array (3 items): [1, 2, 3], array (3 items): [4, 5, 6]]
c: 2
d: 3
flat: array (6 items): [1, 2, 3, 4, …]
dimension: 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. This example has two channels and three positions; the source can use different counts.

Source: Inline notation