The Modern Mathematics of Deep Learning

A finite inverse minimum

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

Choose the network parameter that minimizes squared mismatch after applying the forward operator.

This example shows the source’s optimization-based inverse reconstruction.

Source formula example

\min_{\theta\in{\mathcal{P}}}\|K\Phi(\xi,\theta)-x\|^{2}

exact · This chosen linear model has a closed-form solution; other inverse problems may not.

Symbols

$K$: Forward map; toy K(y) = 2y.
$Φ$: Parameter-dependent candidate; toy Φ(ξ,θ) = ξθ.
$ξ$: Fixed input, here 1.
$θ$: Real parameter.
$x$: Observation.

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.

Observed value

This example’s data flow

You choose

input.observed = 3

Initial values in code

xi = 1

Calculated here

Result expression: { theta, candidate, residual, minimum: residual ** 2 }

observed = input.observed Uses: input.observed
theta = observed / 2 Uses: observed
candidate = xi * theta Uses: xi, theta
residual = 2 * candidate - observed Uses: candidate, observed

Returns

Example result

theta: 1.5
candidate: 1.5
residual: 0
minimum: 0

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Here the parameter domain is all real numbers, so θ = x/2 is the exact minimizer and the minimum is zero.

Source: Source expression