The Modern Mathematics of Deep Learning

A product of hidden widths

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: 3.2 Simple pieces, many layers

Multiply the widths of all hidden layers. The product symbol Π means repeated multiplication.

This product appears in the source discussion of how the number of piecewise-affine regions can scale.

Source formula example

\prod_{\ell=1}^{L-1}N_{\ell}

exact · This computes the product for two widths. It is not an exact count of affine regions for an arbitrary network.

Symbols

$N_ℓ$: Positive integer width of hidden layer ℓ.
$L$: Number of parameterized layers.

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 hidden widthSecond hidden width

This example’s data flow

You choose

input.first = 3
input.second = 4

Calculated here

Result expression: { widths, product: widths.reduce((total, width) => total * width, 1) }

widths = [input.first, input.second] Uses: input.first, input.second

Returns

Example result

widths: array (2 items): [3, 4]
product: 12

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: Hidden-width product