The Modern Mathematics of Deep Learning

Spaces in an inverse problem

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.

Inverse problems · Describe reconstruction from observations using a forward operator.

This application has its own inputs and assumptions; the penguin measurements are not its experimental data.

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: 8.1 Working back from measurements

The forward map K takes an unknown y in 𝒴 to data x in 𝒳. The inverse problem seeks y from x.

The spaces distinguish the unknown from the observation.

Source formula example

{\mathcal{X}},{\mathcal{Y}}

illustration · This scalar example has an exact inverse; a general inverse problem can be non-unique or unstable.

Symbols

$𝒴$: Space of unknowns.
$𝒳$: Space of observations.
$K$: The forward map.
$y$: An unknown.
$x$: Observed data.

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

y = 2

Calculated here

Result expression: { y, x, reconstructed: x / 3 }

x = K(y) Uses: K, y

Returns

Example result

y: 2
x: 6
reconstructed: 2

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. This scalar example has an exact inverse; a general inverse problem can be non-unique or unstable.

Source: Inline notation