The Modern Mathematics of Deep Learning

A Sobolev norm hypothesis

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: 3.2 Simple pieces, many layers

The theorem restricts the size of the function and its derivatives using a Sobolev norm.

The identically zero function has norm zero for every allowed k and p.

The norm bound specifies the class of functions covered by the approximation theorem.

Source formula example

\|g\|_{W^{k,p}((0,1)^{d})}\leq B

illustration · The norm is exactly zero for the zero function. The code does not calculate a general Sobolev norm or prove the approximation theorem.

Symbols

$g$: A function in the specified Sobolev space.
$B$: A positive norm bound.

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.

Positive bound B

This example’s data flow

You choose

input.B = 1

Initial values in code

value = 0
norm = 0

Calculated here

Result expression: { value, norm, B: input.B, withinBound: norm <= input.B }

Returns

Example result

value: 0
norm: 0
B: 1
withinBound: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source theorem uses positive integers d and k with k≥2, p∈[1,∞], and functions g in W^{k,p}((0,1)ᵈ) whose norm is at most B.

Source: Theorem hypothesis