The Modern Mathematics of Deep Learning

Stochastic process path

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

A stochastic process gives a random state for each time t. The source writes Sₓ,ₜ for a process starting at x and Bₜ for Brownian motion.

A process describes how the state changes over time.

Source formula example

(\mathcal{S}_{x,t})_{t\geq 0}

illustration · These are illustrative values at three times, not a sampled Brownian path or an exact solution of the general SDE.

Symbols

$Sₓ,ₜ$: The random state at time t for an initial point x.
$Bₜ$: Brownian motion.
$i$: Index of a process realization.

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 = 1
times = [0, 0.5, 1]
brownianPath = [0, 0.25, -0.25]

Calculated here

Result expression: { x, times, brownianPath, states }

states = brownianPath.map((b) => x + b) Uses: brownianPath, x

Returns

Example result

x: 1
times: array (3 items): [0, 0.5, 1]
brownianPath: array (3 items): [0, 0.25, -0.25]
states: array (3 items): [1, 1.25, 0.75]

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. These are illustrative values at three times, not a sampled Brownian path or an exact solution of the general SDE.

Source: Inline notation