The Modern Mathematics of Deep Learning

ReLU can pass a shifted input

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.

Prediction rules and layers · Describe or apply prediction rules, layers and parameters.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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: 6.2 Shortcuts between layers

If x + b is positive, subtracting b after ReLU returns x.

This local identity helps explain the role of skip connections.

Source formula example

x={\varrho_{R}}(x+b)-b

exact · The equality under this condition does not establish a general residual-network theorem.

Symbols

$x$: Real input; componentwise in a vector.
$b$: Real shift.
$ϱR$: ReLU function.

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.

x

This example’s data flow

You choose

input.x = 1

Initial values in code

b = 3

Calculated here

Result expression: { x, shifted: Math.max(0, x + b) - b, condition: x > -b }

x = input.x Uses: input.x

Returns

Example result

x: 1
shifted: 1
condition: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

For every component the condition is xᵢ > −bᵢ. The toy chooses b = 3 and restricts x to ensure it.

Source: Source expression