The Modern Mathematics of Deep Learning

Coordinate basis vector

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.

Vectors and matrices · Combine numbers using vector and matrix operations.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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? · 6.3 Taking apart and rebuilding

The i-th coordinate basis vector has 1 at index i and 0 at all other positions.

These vectors select individual coordinates.

Source formula example

e^{(i)}\in{\mathbb{R}}^{m}

illustration · The source also uses arbitrary finite index sets G; the example uses positions 1 through 3.

Symbols

$e⁽ⁱ⁾$: The i-th coordinate basis vector.
$i$: A valid one-based index.
$m$: Vector dimension.
$G$: An index set in the source generalization.

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

m = 3
i = 2

Calculated here

Result expression: { m, i, e }

e = Array.from({ length: m }, (_, j) => (j + 1 === i ? 1 : 0)) Uses: m, i

Returns

Example result

m: 3
i: 2
e: array (3 items): [0, 1, 0]

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. The source also uses arbitrary finite index sets G; the example uses positions 1 through 3.

Source: Inline notation