The Modern Mathematics of Deep Learning

A hyperplane region bound

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.

Approximation and depth · Study approximation error and network size under stated conditions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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

The source bounds the number of regions created by n hyperplanes in d dimensions by this sum. The binomial coefficient counts ways to choose j items from n.

The bound limits how many affine regions a shallow ReLU network can create.

Source formula example

\sum_{j=0}^{d}\binom{n}{j}

illustration · This evaluates an upper bound, not the actual count for a particular arrangement or a proof. Degenerate or coincident hyperplanes can produce fewer regions.

Symbols

$n$: Positive number of hyperplanes.
$d$: Positive input dimension.
$\binom nj$: Number of j-element selections from n items; zero if j>n.

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.

Hyperplanes nDimensions d

This example’s data flow

You choose

input.n = 3
input.d = 2

Calculated here

Result expression: { terms, regionBound: terms.reduce((a, b) => a + b, 0) }

n = input.n Uses: input.n
d = input.d Uses: input.d
terms = Array.from({ length: d + 1 }, (_, j) => choose(n, j)) Uses: d, choose, n

Returns

Example result

terms: array (3 items): [1, 3, 3]
regionBound: 7

Some runtime paths or values are not resolved here.

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: Hyperplane region bound