The Modern Mathematics of Deep Learning

Indices outside an initial range

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.

Sets and notation · Describe domains, indices and membership.

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

The set difference [n] ∖ [n/2] contains indices in [n] that are not in the first half, when n is even.

This selects the remaining indices.

Source formula example

j\in[n]\setminus[n/2]

illustration · The example uses an even n, as needed for this integer index range.

Symbols

$n$: A positive even integer.
$j$: An index.
$[n]$: The integers 1 through n.
$[n/2]$: The integers 1 through n/2.

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

n = 6

Calculated here

Result expression: { n, selected: all.filter((j) => j > n / 2) }

all = Array.from({ length: n }, (_, i) => i + 1) Uses: n

Returns

Example result

n: 6
selected: array (3 items): [4, 5, 6]

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 uses an even n, as needed for this integer index range.

Source: Inline notation