The Modern Mathematics of Deep Learning

Spin-block determinants

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

The model multiplies one determinant for each spin block. Swapping two columns of one block reverses its sign.

These expressions specify the inverse or quantum model.

Source formula example

\det\left(\left(\varphi_{i}(r_{j})\right)_{i,j=1}^{n_{+}}\right)\cdot\det\left(\left(\varphi_{i}(r_{j})\right)_{i,j=n_{+}+1}^{n}\right)

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

Symbols

$φᵢ$: Orbital function indexed by i.
$rⱼ$: Position of particle j.
$n₊,n$: Positive-spin particle count and total count.
$det$: Determinant of the indicated spin-block matrix.

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.

Matrix entry

This example’s data flow

You choose

input.x = 2

Initial values in code

second = 2

Calculated here

Result expression: { first, second, product: first * second, swappedProduct: swapped * second }

first = determinant(1, input.x, 3, 4) Uses: determinant, input.x
swapped = determinant(input.x, 1, 4, 3) Uses: determinant, input.x

Returns

Example result

first: -2
second: 2
product: -4
swappedProduct: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented first block is 2 by 2 and the second is 1 by 1. Actual entries are orbital values at particle positions.

Source: Source expression