The Modern Mathematics of Deep Learning

Smooth local functions

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 source uses local coordinates, multiplies the local function by a partition weight, and extends the result to a larger domain. The notation Cᵏ means derivatives through order k are continuous.

The toy uses an identity chart, g(x)=x², and weight t; the local expression t·x² extends to all real x and t.

Smooth local descriptions let the source invoke classical approximation results.

Source formula example

g\circ\gamma_{i}^{-1}\in C^{k}(\gamma_{i}(\mathcal{M}\cap U_{i}))

illustration · The toy verifies explicit polynomial derivatives through order two. It does not prove a general smooth extension theorem or construct a manifold partition of unity.

Symbols

$γ_i$: A local coordinate chart with inverse on its image.
$C^k$: Functions with continuous derivatives through order k.
$t$: The partition weight, in [0,1] in the source local construction.

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.

Local coordinate xWeight t

This example’s data flow

You choose

input.x = 1
input.t = 0.5

Calculated here

Result expression: { localFunction, weighted, derivativeX: 2 * t * x, derivativeT: x ** 2, secondXX: 2 * t, secondXT: 2 * x, secondTT: 0, }

x = input.x Uses: input.x
t = input.t Uses: input.t
localFunction = x ** 2 Uses: x
weighted = t * localFunction Uses: t, localFunction

Returns

Example result

localFunction: 1
weighted: 0.5
derivativeX: 1
derivativeT: 1
secondXX: 1
secondXT: 2
secondTT: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source assumes a compact embedded manifold with local charts and g in Cᵏ on the manifold. Its weighted local functions are Cᵏ on the corresponding chart domain times [0,1], then are extended using an extension theorem. The identity-chart polynomial demonstrates k=2 only.

Source: Source expression