The Modern Mathematics of Deep Learning

Depth-separation size conditions

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: 3.1 A rule based on distance

The theorem assumes d ≥ C and bounds architecture width by Cd⁵ and shallow width by ceᶜᵈ, for positive constants c and C.

These assumptions state the architecture sizes in the separation result.

Source formula example

d\geq C

illustration · The constants here are chosen solely to illustrate the inequalities; they are not the unspecified theorem constants.

Symbols

$d$: Input dimension.
$N$: Architecture widths.
$n$: Shallow network width.
$c,C$: Positive theorem constants.

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

d = 2
C = 1
c = 1
width = 4
n = 3

Calculated here

Result expression: { d, C, c, width, n, dimensionValid: d >= C, widthBound: C * d ** 5, shallowBound: c * Math.exp(c * d), widthValid: width <= C * d ** 5, shallowValid: n <= c * Math.exp(c * d), }

Returns

Example result

d: 2
C: 1
c: 1
width: 4
n: 3
dimensionValid: true
widthBound: 32
shallowBound: ≈7.389

… (10 items)

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 constants here are chosen solely to illustrate the inequalities; they are not the unspecified theorem constants.

Source: Inline notation