The Modern Mathematics of Deep Learning

A hyperplane

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.

Functions and derivatives · Describe maps and how their outputs change.

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: 3.3 Other ways to measure possibility

A point is in the set when its dot product with w plus b equals zero.

The zeros of an affine function form a boundary when w is nonzero.

Source formula example

\{x\in{\mathbb{R}}^{d}\colon\langle x,w\rangle+b=0\}.

exact · The toy uses exactly representable integer coordinates. Exact equality can be unreliable for general floating-point calculations.

Symbols

$w$: A real d-vector; nonzero for a proper hyperplane.
$b$: A real bias.

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.

First coordinate

This example’s data flow

You choose

input.first = 1

Initial values in code

w = [1, -1]
b = 0

Calculated here

Result expression: { score, onBoundary: score === 0 }

x = [input.first, 1] Uses: input.first
score = x.reduce((sum, value, i) => sum + value * w[i], b) Uses: x, w, b

Returns

Example result

score: 0
onBoundary: true

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: Source expression