The Modern Mathematics of Deep Learning

Sobolev spaces

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.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 1.1 A note on symbols · 3.2 Simple pieces, many layers

Sobolev spaces constrain both a function and its derivatives in an Lp sense. At s = 0, W⁰,ᵖ is Lᵖ.

They express smoothness as well as function size.

Source formula example

W^{s,p}({\mathcal{X}})

illustration · This quadratic illustrates derivatives only. General Sobolev spaces use weak derivatives and fractional orders require additional definitions.

Symbols

$𝒳$: A domain in ℝᵈ.
$s$: Nonnegative smoothness order.
$p$: An exponent from 1 to infinity.

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.

Value

This example’s data flow

You choose

input.x = 2

Calculated here

Result expression: { value: x * x, firstDerivative: 2 * x, secondDerivative: 2 }

x = input.x Uses: input.x

Returns

Example result

value: 4
firstDerivative: 4
secondDerivative: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values in this example are invented. This quadratic illustrates derivatives only. General Sobolev spaces use weak derivatives and fractional orders require additional definitions.

Source: Inline notation