The Modern Mathematics of Deep Learning

Composed threshold maps

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.

Frames and sparse signals · Encode or reconstruct signals with frames and dictionaries.

The project scales bill and flipper measurements using training-row means and scales; its feature vector has two entries.

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: 7.2 A few building blocks at a time

Apply one threshold map, then apply another to its output.

These computations organize features and sparse coefficients.

Source formula example

\phi^{(\ell)}\approx(\mathcal{T}_{D^{(\ell)},b^{(\ell)}}\circ\dots\circ\mathcal{T}_{D^{(1)},b^{(1)}})(\tilde{x}),\quad\ell\in[L].

illustration · An example with zero difference does not establish recovery or a general approximation guarantee.

Symbols

$Tᴰ,ᵇ$: Threshold map for dictionary D and threshold b.
$x̃$: Observed input, potentially perturbed.
$φ⁽ˡ⁾$: Layer-ℓ coefficients.
$≈$: Approximation, not guaranteed equality.

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.

Input

This example’s data flow

You choose

input.x = 3

Initial values in code

referenceCoefficient = 1.5

Calculated here

Result expression: { first, second, referenceCoefficient, difference: second - referenceCoefficient, }

first = threshold(input.x, 1) Uses: threshold, input.x
second = threshold(first, 0.5) Uses: threshold, first

Returns

Example result

first: 2
second: 1.5
referenceCoefficient: 1.5
difference: 0

JavaScript

Read numbers from input and return the result. JavaScript basics.

Result

Enable JavaScript to run this example.

Technical statement

Both dictionaries are scalar identities in this invented example. The fixed reference coefficient is 1.5; compare it to the composed output as the input changes. The source uses an approximate recovery relation under its model assumptions.

Source: Source expression