The Modern Mathematics of Deep Learning

Feature norm comparison

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: 6.5 Removing connections

Compare the L² norm under a probability measure with a bound using a supremum on the input ball.

These quantities occur in the pruning argument.

Source formula example

\|f_{i}\|_{L^{2}(\mu)}\leq\|w^{(2)}\|_{1}\|f_{i}\|_{L^{\infty}(B_{1}(0))}\leq\|w^{(2)}\|_{1}.

illustration · This only checks the printed comparison at weight norm 1; it does not repair or prove a possible source typo.

Symbols

$fᵢ$: The rescaled hidden feature defined in (6.10).
$μ$: Probability measure supported on the unit ball.
$\Vert ·\Vert L²$: Square root of the integral of the squared function.
$\Vert ·\Vert L∞$: Essential supremum on the input ball.
$\Vert w⁽²⁾\Vert ₁$: Sum of absolute output weights.

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 = [0, 1]
supremum = 1
weightNorm = 1

Calculated here

Result expression: { l2, supremum, weightNorm, middle: weightNorm * supremum, right: weightNorm }

l2 = Math.sqrt((values[0] ** 2 + values[1] ** 2) / 2) Uses: values

Returns

Example result

l2: ≈0.7071
supremum: 1
weightNorm: 1
middle: 1
right: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy takes μ equally on 0 and 1, weight norm 1, and f(x) = ReLU(x). Its supremum on the entire interval [−1,1] is exactly 1. The source prints an extra weight norm multiplying the L∞ norm of the already rescaled fᵢ; this may be a typographical inconsistency, so this numerical example does not claim that printed middle inequality holds for every weight norm.

Source: Source expression