The Modern Mathematics of Deep Learning

Continuous function space and dual

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.

Approximation and depth · Study approximation error and network size under stated conditions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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?

C(K) is the space of continuous real functions on K. Its continuous dual contains continuous linear functionals on those functions.

The dual is used in the source argument about approximation.

Source formula example

\mathcal{S}

illustration · The example uses two points and a finite linear functional; it does not prove the general representation theorem.

Symbols

$𝒮$: The normed space, here C(K).
$𝒮^{\prime}$: Its continuous dual.
$K$: A compact set.

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

values = [2, 3]
weights = [1, -1]

Calculated here

Result expression: { values, weights, functional }

functional = values.reduce((s, v, i) => s + v * weights[i], 0) Uses: values, weights

Returns

Example result

values: array (2 items): [2, 3]
weights: array (2 items): [1, -1]
functional: -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. The example uses two points and a finite linear functional; it does not prove the general representation theorem.

Source: Inline notation