The Modern Mathematics of Deep Learning

Layer indices and depth

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.

Model classes and structure · Describe families of rules, their size and structural properties.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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: 1.3 The questions ahead · 2.3 The path matters · 3.3 Other ways to measure possibility

L is the number of network layers after the input. ℓ is a layer index within the relevant range.

Indices identify the layer whose activation or weights are discussed.

Source formula example

\ell

illustration · This example has three layers and two hidden layers.

Symbols

$L$: Network depth.
$ℓ$: A layer index.

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

L = 3

Calculated here

Result expression: { L, layers, hiddenLayers: layers.slice(0, -1) }

layers = Array.from({ length: L }, (_, i) => i + 1) Uses: L

Returns

Example result

L: 3
layers: array (3 items): [1, 2, 3]
hiddenLayers: array (2 items): [1, 2]

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 three layers and two hidden layers.

Source: Inline notation