The Modern Mathematics of Deep Learning

Determinants with learned factors

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

Each orbital matrix entry is multiplied by a particle-dependent factor before taking the determinant.

Learned particle factors enrich the determinant while retaining a structure designed for exchange symmetry.

Source formula example

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

illustration · This evaluates both finite spin blocks but does not establish equivariance or model molecular energy.

Symbols

$φᵢ$: Orbital function i.
$rⱼ$: Particle position j.
$Ψⱼ$: Learned factor for particle j.
$θⱼ$: Parameters of factor j.
$n₊,n$: Positive-spin particle count and total count.

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 factor

This example’s data flow

You choose

input.factor = 1

Initial values in code

orbitals = [ [1, 2], [3, 4], ]

Calculated here

Result expression: { firstBlock: matrix, firstDeterminant: first, secondBlock: [[second]], product: first * second, }

factors = [input.factor, 2] Uses: input.factor
matrix = orbitals.map((row) => row.map((v, j) => v * factors[j])) Uses: orbitals, factors
first = matrix[0][0] * matrix[1][1] - matrix[0][1] * matrix[1][0] Uses: matrix
second = 3 * 4

Returns

Example result

firstBlock: array (2 items): [array (2 items): [1, 4], array (2 items): [3, 8]]
firstDeterminant: -4
secondBlock: array (1 items): [array (1 items): [12]]
product: -48

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented positive-spin block is 2 by 2 and the other spin block is 1 by 1, with orbital value 3 and learned factor 4. Particle-indexed learned factors must satisfy the source equivariance condition for the product to have the required antisymmetry.

Source: Source expression