The Modern Mathematics of Deep Learning

Convolutional dictionary

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

Each output channel is the sum of convolutions of dictionary filters with coefficient channels.

These computations organize features and sparse coefficients.

Source formula example

x_{i}=\sum_{j=1}^{C}\kappa_{i,j}\ast\phi_{j},\quad i\in[c],

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$xᵢ$: Output signal in channel i.
$κᵢ,ⱼ$: Convolution filter from coefficient channel j to output channel i.
$φⱼ$: Coefficient signal in channel j.
$C,c$: Numbers of coefficient and output channels.

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.

Second channel value

This example’s data flow

You choose

input.coefficient = 3

Initial values in code

filters = [ [1, 0], [0, 1], ]

Calculated here

Result expression: { result }

phi = [ [1, 2], [input.coefficient, 0], ] Uses: input.coefficient
result = [0, 1].map((i) => filters.reduce( (sum, f, j) => sum + f.reduce((a, v, k) => a + v * phi[j][(i - k + 2) % 2], 0), 0, ), ) Uses: filters, phi

Returns

Example result

result: array (2 items): [1, 5]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numbers in this example are invented.

Source: Source expression