The Modern Mathematics of Deep Learning

Rank-to-sample ratio

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.

Generalization and risk bounds · Relate sample behavior to population risk under stated assumptions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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: 2.4 A second descent

The ratio k*/m compares the effective rank index k* with the training sample size m.

The source requires this ratio to be sufficiently small in its variance estimate.

Source formula example

k^{*}/m

illustration · The example supplies k* directly; it does not calculate it from eigenvalues or test the source asymptotic condition.

Symbols

$k*$: The source effective-rank index.
$m$: Positive training sample size.

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

k = 2
m = 20

Calculated here

Result expression: { k, m, ratio: k / m }

Returns

Example result

k: 2
m: 20
ratio: 0.1

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 example supplies k* directly; it does not calculate it from eigenvalues or test the source asymptotic condition.

Source: Inline notation