The Modern Mathematics of Deep Learning

Regularized 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

Apply the forward map K to a candidate y. Compare the result with observation x, then add a penalty R(y) weighted by α.

The penalty can help choose among candidates that fit observed data.

Source formula example

\|Ky-x\|^{2}+\alpha R(y),

illustration · This evaluates one scalar toy objective. It does not solve a general inverse problem.

Symbols

$K$: A forward map from candidate space to observation space.
$x$: Observed data.
$y$: A candidate unknown.
$R$: A chosen regularizer.
$α$: Penalty 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.

Candidate yPenalty α

This example’s data flow

You choose

input.candidate = 1
input.alpha = 0.5

Calculated here

Result expression: { predictedObservation, residual, fit, penalty, objective: fit + penalty }

predictedObservation = 2 * input.candidate Uses: input.candidate
residual = predictedObservation - 4 Uses: predictedObservation
fit = residual ** 2 Uses: residual
penalty = input.alpha * input.candidate ** 2 Uses: input.alpha, input.candidate

Returns

Example result

predictedObservation: 2
residual: -2
fit: 4
penalty: 0.5
objective: 4.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Change the input and run the same code again. The displayed result uses the values shown above.

Source: Source expression