The Modern Mathematics of Deep Learning

Fixed features

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: 5.1 The shape of training loss · 5.2 When parameters move only a little

Once hidden-unit features are fixed, changing output weights selects a function from their linear span.

These quantities connect network parameters to loss and optimization.

Source formula example

\widetilde{{\mathcal{F}}}_{\theta^{(1)}}\coloneqq\{\Phi(\cdot,\theta)\colon\theta=(\theta^{(1)},\theta^{(2)}),\ \theta^{(2)}\in{\mathbb{R}}^{n}\}

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$θ⁽¹⁾$: Fixed hidden-unit weights and biases.
$θ⁽²⁾$: Real output-weight vector of length n.
$F̃$: Set of network functions produced as output weights vary.

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.

First output weight

This example’s data flow

You choose

input.weight = 2

Initial values in code

features = [2, 1]

Calculated here

Result expression: { features, output: input.weight * features[0] + 3 * features[1] }

Returns

Example result

features: array (2 items): [2, 1]
output: 7

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

The numbers in this example are invented.

Source: Source expression