The Modern Mathematics of Deep Learning

Separate spin-group embeddings

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

Compute distance-based embeddings with separate parameters for positive and negative spin groups.

This example makes the source notation or condition concrete.

Source formula example

\theta_{\textrm{emb}}^{-}

illustration · This finite example is not a trained molecular model and does not prove the equivariance of the complete architecture.

Symbols

$θemb⁻$: Parameters used by negative-spin embedding.
$θemb⁺$: Parameters used by positive-spin embedding.
$Υ$: Shared distance function within each spin group.

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.

Negative-spin scale

This example’s data flow

You choose

input.negativeScale = 2

Initial values in code

positive = [0, 2]
negative = [4, 7]
positiveScale = 1

Calculated here

Result expression: { positiveScale, negativeScale, positiveEmbedding: embed(positive, positiveScale), negativeEmbedding: embed(negative, negativeScale), }

negativeScale = input.negativeScale Uses: input.negativeScale

Returns

Example result

positiveScale: 1
negativeScale: 2
positiveEmbedding: array (2 items): [2, 2]
negativeEmbedding: array (2 items): [6, 6]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented one-dimensional Υ(distance,θ)=θ×distance uses separate scales for the two spin groups. The source builds the negative-spin embedding analogously with its own parameters.

Source: Source expression