The Modern Mathematics of Deep Learning

The spin-glass parameter sphere

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.

Norms and distances · Measure size or difference in a specified space.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 5.1 The shape of training loss

An n-dimensional coordinate vector can lie on a sphere of radius √n; that sphere has dimension n−1.

This example explains the corresponding source notation.

Source formula example

(n-1)

illustration · A point on the sphere does not calculate the Hamiltonian or establish its critical-point statistics.

Symbols

$n$: Positive integer number of ambient coordinates.
$n−1$: Dimension of the sphere.
$\surd n$: Sphere radius.

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.

Coordinate count n

This example’s data flow

You choose

input.n = 4

Calculated here

Result expression: { n, sphereDimension: n - 1, radius, point, norm }

n = input.n Uses: input.n
radius = Math.sqrt(n) Uses: n
point = Array.from({ length: n }, (_, i) => (i === 0 ? radius : 0)) Uses: n, radius
norm = Math.hypot(...point) Uses: point

Returns

Example result

n: 4
sphereDimension: 3
radius: 2
point: array (4 items): [2, 0, 0, 0]
norm: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The example constructs one point in ℝⁿ whose Euclidean norm is √n. The source discusses a spin-glass Hamiltonian on this sphere.

Source: Source expression