The Modern Mathematics of Deep Learning

Mapping inputs to basis vectors

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: 1.2.2 Can the network express the answer?

The source chooses g so that input x⁽ⁱ⁾ maps to the i-th coordinate basis vector e⁽ⁱ⁾.

This construction supports interpolation of distinct training inputs.

Source formula example

g(x^{(i)})=e^{(i)}

illustration · This lists two prescribed input-output pairs; it does not construct the continuous extension or prove the general interpolation theorem.

Symbols

$g$: A vector-valued function.
$x⁽ⁱ⁾$: Training input i.
$e⁽ⁱ⁾$: The i-th coordinate basis vector.

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

inputs = [0, 1]
outputs = [ [1, 0], [0, 1], ]

Calculated here

Result expression: { inputs, outputs }

Returns

Example result

inputs: array (2 items): [0, 1]
outputs: array (2 items): [array (2 items): [1, 0], array (2 items): [0, 1]]

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 lists two prescribed input-output pairs; it does not construct the continuous extension or prove the general interpolation theorem.

Source: Inline notation