The Modern Mathematics of Deep Learning

Support of a local weight

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.

Functions and derivatives · Describe maps and how their outputs change.

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: 4.1 When data live on a smaller shape

The support records where a function can be nonzero, including its boundary in a general continuous setting. The source chooses each partition weight to be supported inside a chart neighborhood.

The finite toy has three points and nonzero weight only at the middle one.

Local support keeps each chart’s contribution in the region where that chart applies.

Source formula example

\supp(\phi_{i})\subset U_{i}

illustration · This checks a finite set, not support closure or an open neighborhood in a continuous manifold. It does not construct a smooth partition of unity.

Symbols

$φ_i$: A partition-of-unity weight.
$U_i$: A chart neighborhood.

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

points = [-1, 0, 1]
values = [0, 1, 0]
neighborhood = [0, 1]

Calculated here

Result expression: { points, values, support, neighborhood, contained: support.every((point) => neighborhood.includes(point)), }

support = points.filter((_, i) => values[i] !== 0) Uses: points, values

Returns

Example result

points: array (3 items): [-1, 0, 1]
values: array (3 items): [0, 1, 0]
support: array (1 items): [0]
neighborhood: array (2 items): [0, 1]
contained: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression