The Modern Mathematics of Deep Learning

A nonzero linear functional

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

Apply a linear functional to an activated affine function and check whether the result is nonzero.

The source uses this condition in its universal approximation argument.

Source formula example

F(\varrho(\langle w,\cdot\rangle+b))\neq 0.

illustration · One nonzero functional and one parameter choice do not prove the condition for every nonzero continuous linear functional.

Symbols

$F$: A nonzero continuous linear functional on the source function space.
$ϱ$: Activation function.
$w,b$: Real weight vector and bias.
$⟨w,·⟩$: Linear function of the input.

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

This example’s data flow

You choose

input.weight = 1

Initial values in code

b = 0

Calculated here

Result expression: { hAtOne, functionalValue, nonzero: functionalValue !== 0 }

w = input.weight Uses: input.weight
hAtOne = Math.max(0, w * 1 + b) Uses: w, b
functionalValue = hAtOne Uses: hAtOne

Returns

Example result

hAtOne: 1
functionalValue: 1
nonzero: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy function space is continuous real functions on [0,1] with the sup norm, F(h)=h(1), and ϱ is ReLU. Evaluation at 1 is a nonzero continuous linear functional. The source uses the condition that for every nonzero continuous linear functional there exist w and b making this value nonzero; that quantified statement underpins its density argument.

Source: Source expression