The Modern Mathematics of Deep Learning

Duplicate features and interpolation

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.

Optimization theory · Study convergence and loss geometry under assumptions.

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: 5.1 The shape of training loss

If two hidden units have the same feature, moving weight from one to the other while keeping their sum fixed preserves the output.

These quantities connect network parameters to loss and optimization.

Source formula example

\Phi(\cdot,(\theta_{*}^{(1)},\theta^{(2)}))=\Phi(\cdot,(\theta_{*}^{(1)},\lambda\theta_{*}^{(2)}+(1-\lambda)\theta^{(2)})),\quad\lambda\in[0,1].

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

Symbols

$λ$: Real interpolation parameter in [0,1].
$θ⁽¹⁾$: Fixed incoming parameters of the hidden units.
$θ⁽²⁾$: Real output-weight vector.
$Φ$: Network 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.

Interpolation

This example’s data flow

You choose

input.lambda = 0.5

Calculated here

Result expression: { weights, output: weights[0] * 2 + weights[1] * 2 }

a = input.lambda Uses: input.lambda
weights = [a, 1 - a] Uses: a

Returns

Example result

weights: array (2 items): [0.5, 0.5]
output: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The equality holds here because both features equal 2 and the two weights always sum to 1. The source path uses duplicated units and requires its stated width and construction assumptions.

Source: Source expression