The Modern Mathematics of Deep Learning

Inverse problem objective

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

Compare the forward prediction Ky to observed data x, square the error, and add a weighted penalty.

These expressions specify the inverse or quantum model.

Source formula example

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

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$K$: Forward operator mapping candidate y to observations.
$x$: Observed data.
$y$: Candidate unknown.
$R$: Regularization function.
$α$: Nonnegative regularization 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.

CandidatePenalty weight

This example’s data flow

You choose

input.y = 1
input.alpha = 0.5

Initial values in code

observed = 3

Calculated here

Result expression: { predicted, observed, matchesObservation: predicted === observed, error, penalty, objective: error ** 2 + penalty, }

predicted = 2 * input.y Uses: input.y
error = predicted - observed Uses: predicted, observed
penalty = input.alpha * input.y ** 2 Uses: input.alpha, input.y

Returns

Example result

predicted: 2
observed: 3
matchesObservation: false
error: -1
penalty: 0.5
objective: 1.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented scalar forward map is K(y) = 2y and the penalty is R(y) = y². Evaluating one candidate does not solve the inverse problem.

Source: Source expression