The Modern Mathematics of Deep Learning

Directions in the source figure

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: 3.1 A rule based on distance

The source figure uses four two-dimensional weight directions. Putting each direction into a row gives the first-layer weight matrix.

The weight directions determine the ridge-function frequency supports in the figure.

Source formula example

W^{(1)}=[w_1\ \cdots\ w_4]^T

illustration · The directions are invented for this example; only the matrix shape and source relationship are taken from the figure.

Symbols

$w_j$: A two-dimensional weight direction.
$W^{(1)}$: A 4-by-2 first-layer weight matrix.

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 direction’s first entry

This example’s data flow

You choose

input.first = 1

Calculated here

Result expression: { directions, matrix, rows: matrix.length, columns: matrix[0].length }

directions = [ [input.first, 0], [0, 1], [-1, 0], [0, -1], ] Uses: input.first
matrix = directions.map((direction) => [...direction]) Uses: directions

Returns

Example result

directions: array (4 items): [array (2 items): [1, 0], array (2 items): [0, 1], array (2 items): [-1, 0], array (2 items): [0, -1]]
matrix: array (4 items): [array (2 items): [1, 0], array (2 items): [0, 1], array (2 items): [-1, 0], array (2 items): [0, -1]]
rows: 4
columns: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Direction matrix in figure