The Modern Mathematics of Deep Learning

A radial function

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: 3.1 A rule based on distance

First find the sum of squares of the entries of x. Apply the one-variable function g-tilde to that number.

This example chooses g-tilde(t) = t, so g(x) is the squared length.

The value of a radial function depends on the vector through its distance from the origin.

Source formula example

g(x)=\tilde{g}(\|x\|^{2}_{2})

exact · The example uses one simple choice of g-tilde and two coordinates. It does not establish a depth-separation result.

Symbols

$x$: Real vector in ℝᵈ.
$\widetilde g$: A function from nonnegative real numbers to real numbers.

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 = 3

Calculated here

Result expression: { squaredLength, value: squaredLength }

x = [input.first, 4] Uses: input.first
squaredLength = x.reduce((sum, value) => sum + value * value, 0) Uses: x

Returns

Example result

squaredLength: 25
value: 25

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