The Modern Mathematics of Deep Learning

Vector-valued data samples

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: 5.2 When parameters move only a little

A sample is an ordered sequence of vectors or input-target pairs, with each vector having the stated dimension.

Shapes show which data can be supplied to an operation.

Source formula example

{s}=((x^{(i)},y^{(i)}))_{i=1}^{m}\in({\mathbb{R}}^{d}\times{\mathbb{R}})^{m}

illustration · The example uses vector inputs and scalar targets; a batch of vectors has the analogous list shape.

Symbols

$m$: Number of examples.
$x⁽ⁱ⁾$: An input vector.
$y⁽ⁱ⁾$: A target scalar or vector as specified in its source expression.
$d, n$: Vector dimensions.
$b$: A batch of 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

sample = [ [[1, 2], 3], [[3, 4], 5], ]
d = 2

Calculated here

Result expression: { sample, d, m: sample.length, valid: sample.every((z) => z[0].length === d && Number.isFinite(z[1])), }

Returns

Example result

sample: array (2 items): [array (2 items): [array (2 items), 3], array (2 items): [array (2 items), 5]]
d: 2
m: 2
valid: 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. The example uses vector inputs and scalar targets; a batch of vectors has the analogous list shape.

Source: Inline notation