The Modern Mathematics of Deep Learning

Testing an integral condition

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: 1.2 Learning from examples

The source proof considers a signed measure μ whose integral against every activated affine function is zero.

This toy evaluates one such integral for a finite signed measure and one choice of w and b. A finite signed measure can have negative weights.

The proof uses this condition to study whether the network functions are dense.

Source formula example

\int_{K}\varrho(\langle w,x\rangle+b)\,\mathrm{d}\mu(x)=0

illustration · A zero result for one w and b does not establish the condition for every w and b. The toy is not a proof of universal approximation.

Symbols

$K$: A compact subset of ℝᵈ.
$μ$: A signed Borel measure on K.
$ρ$: An activation function; the toy uses ReLU.

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.

Weight wBias b

This example’s data flow

You choose

input.w = 1
input.b = 0

Initial values in code

points = [0, 1, 2]
weights = [1, -2, 1]

Calculated here

Result expression: { points, weights, activated, integral }

activated = points.map((x) => Math.max(0, input.w * x + input.b)) Uses: points, input.w, input.b
integral = activated.reduce((sum, value, i) => sum + value * weights[i], 0) Uses: activated, weights

Returns

Example result

points: array (3 items): [0, 1, 2]
weights: array (3 items): [1, -2, 1]
activated: array (3 items): [0, 1, 2]
integral: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source describes the discriminatory condition: if a signed Borel measure gives zero integrals for every w and b, then that measure must be zero. The example only evaluates a single choice and a nonzero finite signed measure. For w = 1 and b = −1, this measure gives integral 1, so it fails the all-parameters zero condition.

Source: Source expression