The Modern Mathematics of Deep Learning

Local kernel support

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

A kernel is local when its nonzero positions are contained in its window.

This calculation explains the corresponding source expression.

Source formula example

\supp(\kappa_{i,j})\subset H

illustration · This single kernel does not prove local sparse recovery.

Symbols

$supp(κᵢ,ⱼ)$: Set of positions where the kernel is nonzero.
$H$: Allowed window positions.
$G$: Finite cyclic position group.

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 kernel entry

This example’s data flow

You choose

input.first = 1

Initial values in code

window = [0, 1]

Calculated here

Result expression: { kernel, window, support, contained: support.every((i) => window.includes(i)) }

kernel = [input.first, 2, 0] Uses: input.first
support = kernel.map((v, i) => (v !== 0 ? i : -1)).filter((i) => i >= 0) Uses: kernel

Returns

Example result

kernel: array (3 items): [1, 2, 0]
window: array (2 items): [0, 1]
support: array (2 items): [0, 1]
contained: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy group has d=3 positions and window length n=2. The zero outside the window enforces the displayed support condition. The source uses such kernels for each pair of channels.

Source: Source expression