The Modern Mathematics of Deep Learning

Thresholding coefficients

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

Project onto dictionary columns. Apply ReLU to the positive and negative parts after subtracting the threshold.

Thresholding can set small coefficients to zero.

Source formula example

\mathcal{T}_{D,b}(x)\coloneqq{\varrho_{R}}(D^{T}x-b)-{\varrho_{R}}(-D^{T}x-b),\quad x\in{\mathbb{R}}^{dc},

exact · The result is exact for these finite toy inputs, apart from ordinary computer rounding.

Symbols

$D$: Dictionary matrix; the toy uses the 1-by-1 identity.
$b$: Threshold; this example uses a nonnegative scalar.

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.

ProjectionThreshold

This example’s data flow

You choose

input.x = 2
input.threshold = 1

Calculated here

Result expression: { projection, value: Math.max(0, projection - threshold) - Math.max(0, -projection - threshold), }

projection = input.x Uses: input.x
threshold = input.threshold Uses: input.threshold

Returns

Example result

projection: 2
value: 1

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