The Modern Mathematics of Deep Learning

Distance on a sample

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.

Norms and distances · Measure size or difference in a specified space.

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: 1.2 Learning from examples

Subtract the two function values on each example, square the differences, average them, and take the square root.

The distance compares functions on a particular sample.

Source formula example

d_{S}(f,g)=d_{(Z^{(i)})_{i=1}^{m}}(f,g)\coloneqq\sqrt{\frac{1}{m}\sum_{i=1}^{m}\big(f(Z^{(i)})-g(Z^{(i)})\big)^{2}}.

exact · Functions that agree on the displayed sample can still differ elsewhere.

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 f value

This example’s data flow

You choose

input.first = 1

Initial values in code

g = [0, 0]

Calculated here

Result expression: { differences, distance: Math.sqrt( differences.reduce((sum, x) => sum + x * x, 0) / differences.length, ), }

f = [input.first, 2] Uses: input.first
differences = f.map((value, i) => value - g[i]) Uses: f, g

Returns

Example result

differences: array (2 items): [1, 2]
distance: ≈1.581

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