The Modern Mathematics of Deep Learning

Check the printed threshold 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.

Frames and sparse signals · Encode or reconstruct signals with frames and dictionaries.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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: 7.2 A few building blocks at a time

The printed condition compares ψ with the absolute value of the threshold. Test the condition and the threshold output separately.

The comparison makes a possible source inconsistency visible.

Source formula example

\psi_{i}\leq|b_{i}|

illustration · This counterexample tests the printed condition. It does not determine whether the source intended a different expression.

Symbols

$ψᵢ$: Real projected coefficient.
$bᵢ$: Real threshold, nonnegative in this toy.

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.

Projection

This example’s data flow

You choose

input.psi = -2

Initial values in code

b = 1

Calculated here

Result expression: { printedCondition, output, zeroed: output === 0 }

psi = input.psi Uses: input.psi
printedCondition = psi <= Math.abs(b) Uses: psi, b
output = Math.max(0, psi - b) - Math.max(0, -psi - b) Uses: psi, b

Returns

Example result

printedCondition: true
output: -1
zeroed: false

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

With ψᵢ = −2 and bᵢ = 1, the printed inequality is true but the threshold output is −1, not zero. For a nonnegative threshold the sufficient and necessary condition for zero output is |ψᵢ| ≤ bᵢ.

Source: Source expression