The Modern Mathematics of Deep Learning

A finite Hilbert space example

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: 6.5 Removing connections

A Hilbert space has an inner product and its associated norm. F is a subset of the space, and g is a point in its convex hull in the source statement.

The setting describes functions and their convex combinations.

Source formula example

(\mathcal{H},\|\cdot\|)

illustration · The Euclidean plane is a finite-dimensional Hilbert space; this example does not prove the general approximation result.

Symbols

$ℋ$: A Hilbert space.
$\Vert ·\Vert$: Its norm.
$F$: A bounded subset of ℋ.
$fᵢ$: An element of F.
$g$: An element in the convex hull of F.

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.

This example’s data flow

Initial values in code

F = [ [1, 0], [0, 1], ]
weights = [0.25, 0.75]

Calculated here

Result expression: { F, weights, g, norm: Math.hypot(...g) }

g = [0, 1].map((i) => F.reduce((s, f, j) => s + weights[j] * f[i], 0)) Uses: F, weights

Returns

Example result

F: array (2 items): [array (2 items): [1, 0], array (2 items): [0, 1]]
weights: array (2 items): [0.25, 0.75]
g: array (2 items): [0.25, 0.75]
norm: ≈0.7906

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. The Euclidean plane is a finite-dimensional Hilbert space; this example does not prove the general approximation result.

Source: Inline notation