The Modern Mathematics of Deep Learning

Random iterative approximation

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.

PDE and stochastic processes · Describe differential equations and stochastic representations.

PDE and quantum models are not operations on these penguin measurements.

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: 4.3 Structure from an equation

Yₓ is a random quantity with mean g(x). The maps Tℓ iteratively produce an approximation 𝒴ₓ⁽ᴸ⁾; X is a random input and YX uses that input.

This representation turns function approximation into approximation of a random iterative procedure.

Source formula example

Y_{x}

illustration · This finite law and single iteration illustrate the roles; no convergence or independence claim is established.

Symbols

$Yₓ$: A random quantity depending on input x.
$g$: Its expectation as a function of x.
$Tℓ$: An iteration map.
$𝒴ₓ⁽ᴸ⁾$: The L-step approximation.
$X$: A random input.
$YX$: The corresponding random output.

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

x = 2
outcomes = [1, 3]
probabilities = [0.5, 0.5]
initial = 0

Calculated here

Result expression: { x, outcomes, probabilities, g, initial, approximation: T1(initial) }

g = outcomes.reduce((s, y, i) => s + y * probabilities[i], 0) Uses: outcomes, probabilities

Returns

Example result

x: 2
outcomes: array (2 items): [1, 3]
probabilities: array (2 items): [0.5, 0.5]
g: 2
initial: 0
approximation: 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 unless a source value is specified. This finite law and single iteration illustrate the roles; no convergence or independence claim is established.

Source: Inline notation