The Modern Mathematics of Deep Learning

A sequence of residual states

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.

Residual networks · Describe residual layer updates and their continuous analogue.

Convolution, recurrent and residual architectures are not part of this linear penguin model.

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

Repeated residual layers produce an ordered sequence of outputs, each depending on the input and parameters.

This example makes the source notation or condition concrete.

Source formula example

(\bar{\Phi}^{(\ell)})_{\ell=0}^{L-1}

illustration · The finite scalar recurrence does not establish an ODE limit or a training result.

Symbols

$Φ̄⁽ˡ⁾$: Residual output at layer ℓ, depending on x and θ.
$x$: Input and initial state.
$θ$: Network parameters.
$h$: Activated update term.
$L$: Layer count.

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.

Initial input

This example’s data flow

You choose

input.x = 1

Initial values in code

theta = 1

Calculated here

Result expression: { x, theta, states }

x = input.x Uses: input.x
state = x Uses: x
states = [state] Uses: state
for (let layer = 1; layer < 3; layer++) { const h = Math.max(0, theta * state); state += h; states.push(state); } Uses: theta, state, states

Returns

Example result

x: 1
theta: 1
states: array (3 items): [1, 2, 4]

Some runtime paths or values are not resolved here.

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This toy uses L=3, scalar θ=1, h=ReLU(θ times the previous state), and the sequence for ℓ=0,1,2. The source uses the residual sequence to motivate an ordinary differential equation.

Source: Source expression