The Modern Mathematics of Deep Learning

A two-dimensional network trajectory

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: 3.3 Other ways to measure possibility

The source figure sends a circle in two dimensions through a ReLU network and shows the resulting output curve.

The toy uses two hidden units, identity weights, zero biases and one hidden layer.

Following a whole input curve reveals how a network changes its shape.

Source formula example

t\mapsto\Phi_{((2,n,\ldots,n,2),\varrho_R)}(\gamma(t),\theta)

illustration · This computes one point on an invented network’s curve. The source figure uses randomly initialized networks with different depths; the toy does not reproduce that experiment.

Symbols

$γ$: The input curve γ(t)=(cos 2πt,sin 2πt).
$n$: Hidden-layer width; the toy uses n=2.

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.

Curve parameter t

This example’s data flow

You choose

input.t = 0

Calculated here

Result expression: { point, hidden, output }

t = input.t Uses: input.t
angle = 2 * Math.PI * t Uses: t
point = [Math.cos(angle), Math.sin(angle)] Uses: angle
hidden = point.map((value) => Math.max(0, value)) Uses: point
output = [...hidden] Uses: hidden

Returns

Example result

point: array (2 items): [1, 0]
hidden: array (2 items): [1, 0]
output: array (2 items): [1, 0]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source figure studies a two-dimensional circle and networks with width 20 and several depths. The example chooses width 2 with identity weights to make one layer’s effect directly calculable.

Source: Network trajectory figure