The Modern Mathematics of Deep Learning

Layer weight and bias shapes

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.

Prediction rules and layers · Describe or apply prediction rules, layers and parameters.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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 Learning from examples · 1.2.2 Can the network express the answer?

A layer with d inputs and n outputs has an n-by-d weight matrix and n biases. The parameter pair combines both.

These dimensions make matrix multiplication and bias addition possible.

Source formula example

W^{(\ell)}\in{\mathbb{R}}^{N_{\ell}\times N_{\ell-1}}

illustration · This checks one layer shape; entries and sizes vary between networks.

Symbols

$W⁽ℓ⁾$: A layer weight matrix.
$b⁽ℓ⁾$: A layer bias vector.
$Nℓ$: Number of layer outputs.
$Nℓ₋₁$: Number of layer inputs.
$θ⁽¹⁾$: The first layer weight and bias pair.
$d, n, m$: Positive integer widths in the relevant expressions.

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

W = [ [1, 2], [3, 4], [5, 6], ]
b = [0, 1, 2]
d = 2
n = 3

Calculated here

Result expression: { W, b, rows: W.length, columns: W[0].length, valid: W.length === n && W.every((row) => row.length === d) && b.length === n, }

Returns

Example result

W: array (3 items): [array (2 items): [1, 2], array (2 items): [3, 4], array (2 items): [5, 6]]
b: array (3 items): [0, 1, 2]
rows: 3
columns: 2
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. This checks one layer shape; entries and sizes vary between networks.

Source: Inline notation