The Modern Mathematics of Deep Learning

Duplicated hidden 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

A duplicated hidden unit computes the same feature as an earlier unit for every input.

These quantities connect network parameters to loss and optimization.

Source formula example

\varrho\big(\langle(\theta_{*}^{(1)})_{j},\begin{bmatrix}\cdot\\ 1\end{bmatrix}\rangle\big)\in\left\{\varrho\big(\langle(\theta_{*}^{(1)})_{k},\begin{bmatrix}\cdot\\ 1\end{bmatrix}\rangle\big)\colon k\in[n/2]\right\},

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

Symbols

$θ*⁽¹⁾$: Hidden parameters in the constructed path.
$n$: Even positive number of hidden units.
$j,k$: Hidden-unit indices.
$ϱ$: Activation function.

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.

Input

This example’s data flow

You choose

input.x = 1

Calculated here

Result expression: { first, duplicate }

x = input.x Uses: input.x
first = Math.max(0, 2 * x + 1) Uses: x
duplicate = Math.max(0, 2 * x + 1) Uses: x

Returns

Example result

first: 3
duplicate: 3

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The two invented units use identical weight and bias, so their functions agree for every real input. The source builds a wider path using duplicated features.

Source: Source expression