The Modern Mathematics of Deep Learning

Real vectors

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.

Sets and notation · Describe domains, indices and membership.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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 · 1.2.2 Can the network express the answer? · 2.3 The path matters · More source sections

A vector in ℝᵈ has d real-number entries. The example uses d = 2.

Vector dimensions must match the operation.

Source formula example

{\mathbb{R}}^{p}

illustration · This invented finite example illustrates the notation; it does not establish a general theorem.

Symbols

$x, w, θ$: Real vectors; each occurrence names its vector in context.
$d, p$: The number of entries in the relevant vector.

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.

Value

This example’s data flow

You choose

input.x = 2

Calculated here

Result expression: { x, dimension: x.length }

x = [input.x, 3] Uses: input.x

Returns

Example result

x: array (2 items): [2, 3]
dimension: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values in this example are invented. This invented finite example illustrates the notation; it does not establish a general theorem.

Source: Inline notation