The Modern Mathematics of Deep Learning

One added input coordinate

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: 2.4 A second descent

A d-entry vector gains one entry when its dimension changes from d to d+1.

The added coordinate extends a feature vector.

Source formula example

d\mapsto d+1

illustration · These are realized values, not a simulation of the random input distribution.

Symbols

$d$: Original vector dimension.
$Xᵢ$: The i-th coordinate of the random vector X.
$Xᵈ⁺¹$: The added coordinate.

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

x = [1, -1]
newCoordinate = 1

Calculated here

Result expression: { d, x, newCoordinate, extended: [...x, newCoordinate], newDimension: d + 1 }

d = x.length Uses: x

Returns

Example result

d: 2
x: array (2 items): [1, -1]
newCoordinate: 1
extended: array (3 items): [1, -1, 1]
newDimension: 3

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. These are realized values, not a simulation of the random input distribution.

Source: Inline notation