The Modern Mathematics of Deep Learning

Features and targets from a transformation

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.

Classification predicts three species; Ridge predicts body mass in grams using the same observations. These are separate models. Training predictions feed the objective; 102 separate test predictions never update the weights.

Observations and sampling · Describe examples, targets and samples.

The selected rows supply measurements and known answers to training; held-out IDs remain separate.

TaskTraining rowsTraining stepsPenalty αHeld-out row ID

Features

Input: Training: 12 × 2 · First row #19: 34.4 mm, 184 mm
Output: [≈-1.946, ≈-1.283]

Project operation

The project calculation uses these records; the source formula and example follow below. Code edits run locally; project settings travel between pages.

Observations → Features → Prediction

Project input

Showing first three entries; the operation receives the complete array. Rounded preview; the operation uses full precision.

input.rows · Observations: [Bill length (mm): 34.4 · Flipper length (mm): 184, Bill length (mm): 40.9 · Flipper length (mm): 187, Bill length (mm): 41.7 · Flipper length (mm): 210] … (12 entries)

JavaScript

Result

[[≈-1.946, ≈-1.283], [≈-0.6098, ≈-1.029], [≈-0.4454, ≈0.9162]] … (12 entries)

Used in these source sections: 1.2 Learning from examples

The source describes making a feature T₁(z) and target T₂(z) from an unlabeled item z. When T₁ has an inverse on the relevant domain, the target can be expressed as T₂(T₁⁻¹(x)).

The toy uses T₁(z)=2z and T₂(z)=z+1.

This construction turns transformed data into a prediction task.

Source formula example

T_{2}\circ T_{1}^{-1}\colon{\mathcal{X}}\to{\mathcal{Y}}

exact · The toy uses an invertible scalar feature map. A general transformation need not have an inverse; the source expression requires a well-defined inverse on the relevant domain.

Symbols

$T_1$: A transformation from original data to features.
$T_2$: A transformation from original data to targets.
$T_1^{-1}$: An inverse on the relevant feature domain, when it exists.

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.

Original value z

This example’s data flow

You choose

input.z = 3

Calculated here

Result expression: { feature, target, recovered, composed }

z = input.z Uses: input.z
feature = 2 * z Uses: z
target = z + 1 Uses: z
recovered = feature / 2 Uses: feature
composed = recovered + 1 Uses: recovered

Returns

Example result

feature: 6
target: 4
recovered: 3
composed: 4

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: Transformed prediction