The Modern Mathematics of Deep Learning

Function norms with a measure

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.

Norms and distances · Measure size or difference in a specified space.

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: 4.1 When data live on a smaller shape · 4.2 Approximating with random pieces

The Lᵖ(μ) norm uses μ to weight the integral of absolute function values raised to p.

The measure specifies which parts of the domain contribute to the function norm.

Source formula example

L^{p}(\mu)

illustration · This finite example uses p = 2; p = ∞ uses an essential supremum.

Symbols

$p$: A norm exponent.
$μ$: The measure used in the integral.
$Lᵖ(μ)$: The associated function space.

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 = [1, 3]
weights = [0.5, 0.5]
p = 2

Calculated here

Result expression: { p, norm: Math.sqrt(values.reduce((s, v, i) => s + weights[i] * v * v, 0)) }

Returns

Example result

p: 2
norm: ≈2.236

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. This finite example uses p = 2; p = ∞ uses an essential supremum.

Source: Inline notation