The Modern Mathematics of Deep Learning

Combining local descriptions

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.

Approximation and depth · Study approximation error and network size under stated conditions.

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: 4.1 When data live on a smaller shape

Each chart γᵢ gives local coordinates; its inverse returns the point. The weights φᵢ add to one and select the local descriptions that apply.

This toy uses two identity charts and fixed weights 1/4 and 3/4.

Local coordinates can describe a function on a lower-dimensional manifold.

Source formula example

g(x)=\sum_{i=1}^{p}\phi_{i}(x)\left(g\circ\gamma_{i}^{-1}(\gamma_{i}(x))\right)\eqqcolon\sum_{i=1}^{p}\tilde{g}_{i}(\gamma_{i}(x),\phi_{i}(x)),\quad x\in\mathcal{M},

illustration · The identity charts and weights are a finite toy. This evaluates one point and does not establish an approximation theorem.

Symbols

$ℳ$: The manifold containing x.
$γ_i$: A local coordinate chart with an inverse on its domain.
$φ_i$: A partition-of-unity weight.

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.

Point coordinate

This example’s data flow

You choose

input.x = 2

Initial values in code

weights = [0.25, 0.75]

Calculated here

Result expression: { coordinates, weights, localValues, combined: localValues.reduce((sum, value, i) => sum + value * weights[i], 0), }

x = input.x Uses: input.x
coordinates = [x, x] Uses: x
localValues = coordinates.map((coordinate) => coordinate ** 2) Uses: coordinates

Returns

Example result

coordinates: array (2 items): [2, 2]
weights: array (2 items): [0.25, 0.75]
localValues: array (2 items): [4, 4]
combined: 4

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source combines local charts and a partition of unity on the manifold. Chart inverses are used only where the corresponding chart applies. The toy uses overlapping identity charts and weights that sum to one.

Source: Source expression