The Modern Mathematics of Deep Learning

A high-dimensional size scale

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: 1.3 The questions ahead

The source discusses a size scale with exponent 784, corresponding to the number of pixels in a 28-by-28 image.

To avoid overflowing ordinary computer numbers, the code returns the base-10 logarithm of the factor.

Large exponents make a size factor grow quickly as target error shrinks.

Source formula example

\mathcal O(\varepsilon^{-784})

approximation · The displayed logarithm is subject to floating-point rounding. A null factor means it exceeds this toy’s finite-number range. Big-O notation hides constants; the code does not calculate the number of parameters for a particular network.

Symbols

$ε$: Target uniform approximation error in (0,1).

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.

Accuracy ε

This example’s data flow

You choose

input.epsilon = 0.1

Calculated here

Result expression: { log10Factor, representableFactor }

epsilon = input.epsilon Uses: input.epsilon
log10Factor = -784 * Math.log10(epsilon) Uses: epsilon
representableFactor = log10Factor <= 308 ? 10 ** log10Factor : null Uses: log10Factor

Returns

Example result

log10Factor: 784
representableFactor: null

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source discusses uniform approximation of 1-Lipschitz functions on a 784-dimensional unit cube in the context of worst-case approximation complexity. The computed quantity is the factor ε⁻⁷⁸⁴ and its logarithm, not a measured network size.

Source: High-dimensional scale