The Modern Mathematics of Deep Learning

Permutation-equivariant features

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.

Quantum models · Describe wavefunctions, particle coordinates and energy relations.

PDE and quantum models are not operations on these penguin measurements.

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: 8.2 Equations as guides

When particle positions are swapped, features attached to those particles must swap in the same way.

These expressions specify the inverse or quantum model.

Source formula example

P_{i,j}\Psi_{k}(\cdot,\theta_{k})=\begin{cases}\Psi_{k}(\cdot,\theta_{k}),&\text{if }k\notin\{i,j\},\\ \Psi_{i}(\cdot,\theta_{i}),&\text{if }k=j,\\ \Psi_{j}(\cdot,\theta_{j}),&\text{if }k=i,\end{cases}

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$Pᵢ,ⱼ$: Swap of particle coordinates i and j.
$Ψₖ$: Particle-indexed learned function.
$θₖ$: Parameters of function k.

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.

First position

This example’s data flow

You choose

input.first = 1

Calculated here

Result expression: { features, featuresAfterSwap: swappedPositions.map((x) => x * x) }

positions = [input.first, 3] Uses: input.first
features = positions.map((x) => x * x) Uses: positions
swappedPositions = [positions[1], positions[0]] Uses: positions

Returns

Example result

features: array (2 items): [1, 9]
featuresAfterSwap: array (2 items): [9, 1]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This invented shared feature map is equivariant for the two displayed positions. The source requires the stated permutation rule for all relevant particle swaps.

Source: Source expression