The Modern Mathematics of Deep Learning

Distinct input points

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.

Observations and sampling · Describe examples, targets and samples.

The selected rows supply measurements and known answers to training; held-out IDs remain separate.

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: 1.2.2 Can the network express the answer? · 5.2 When parameters move only a little

The condition x⁽ⁱ⁾ ≠ x⁽ʲ⁾ says the two input values differ.

Distinct training inputs can be required for interpolation.

Source formula example

x^{(i)}\neq x^{(j)}

illustration · This compares one pair of equal-length vectors; a theorem may require every pair of inputs to differ.

Symbols

$x⁽ⁱ⁾, x⁽ʲ⁾$: Two input vectors.

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

first = [1, 2]
second = [1, 3]

Calculated here

Result expression: { first, second, distinct: first.some((v, i) => v !== second[i]) }

Returns

Example result

first: array (2 items): [1, 2]
second: array (2 items): [1, 3]
distinct: 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 compares one pair of equal-length vectors; a theorem may require every pair of inputs to differ.

Source: Inline notation