The Modern Mathematics of Deep Learning

An empirical feature matrix

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.

Vectors and matrices · Combine numbers using vector and matrix operations.

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.

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: 2.4 A second descent

The matrix Σ is the sum of the outer products X⁽ⁱ⁾(X⁽ⁱ⁾)ᵀ over the observed sample.

This matrix combines the training input coordinates.

Source formula example

\Sigma

illustration · This is a sum over four observed vectors. It is not the population covariance or the sample average.

Symbols

$X$: A training input vector.
$Σ$: The sum of sample outer products.

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.

This example’s data flow

Initial values in code

vectors = [ [1, 0], [-1, 0], [0, 1], [0, -1], ]

Calculated here

Result expression: { vectors, sigma }

sigma = [0, 1].map((i) => [0, 1].map((j) => vectors.reduce((s, x) => s + x[i] * x[j], 0)), ) Uses: vectors

Returns

Example result

vectors: array (4 items): [array (2 items): [1, 0], array (2 items): [-1, 0], array (2 items): [0, 1], array (2 items): [0, -1]]
sigma: array (2 items): [array (2 items): [2, 0], array (2 items): [0, 2]]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. This is a sum over four observed vectors. It is not the population covariance or the sample average.

Source: Inline notation