The Modern Mathematics of Deep Learning

Where a unit switches

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

This set contains inputs where the selected unit’s preactivation equals zero. It is the boundary between positive and negative preactivation regions.

The toy preactivation is ReLU(x)−1, which is zero at x=1.

These boundaries partition input space into regions with different activation patterns.

Source formula example

H_{i,\ell}\coloneqq\{x\in{\mathbb{R}}^{N_{0}}\colon\Phi_{i}^{(\ell)}(x,\theta)=0\},

illustration · The code checks membership for a single chosen point. A finite test cannot enumerate a general continuous boundary; exact floating-point equality can be unreliable for other inputs.

Symbols

$H_{i,ℓ}$: Zero set of the preactivation for unit i in layer ℓ.
$N_0$: Input dimension.

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

Calculated here

Result expression: { first, preactivation, onBoundary: preactivation === 0, outsideBoundary: preactivation !== 0, }

first = Math.max(0, input.x) Uses: input.x
preactivation = first - 1 Uses: first

Returns

Example result

first: 1
preactivation: 0
onBoundary: true
outsideBoundary: false

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

In a deep ReLU network, preactivations are piecewise affine, so the global zero set need not be a single hyperplane. The toy has one input and two sequential operations; outsideBoundary checks membership in the complement of this one zero set. The source also considers complements of unions over many units and layers.

Source: Activation boundary