The Modern Mathematics of Deep Learning

Zero-padded window convolution

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.

Convolution · Describe shared filters and downsampling.

Convolution, recurrent and residual architectures are not part of this linear penguin model.

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: 6.3 Taking apart and rebuilding

Extend the short window by zeros to the full circular group, then use circular convolution.

This definition connects window filters and the framelet encoder.

Source formula example

g\ast_{\circ}v\coloneqq g\ast\bar{v}

illustration · This uses a finite cyclic group and one window.

Symbols

$g$: Signal on the finite cyclic group G.
$v$: Window signal on H.
$v̄$: Zero-padded version of v on G.
$∗$: Circular group convolution.
$∗\circ$: Convolution with the padded window.

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 window value

This example’s data flow

You choose

input.first = 1

Initial values in code

g = [1, 2, 3]

Calculated here

Result expression: { g, v, padded, convolution }

v = [input.first, 2] Uses: input.first
padded = [...v, 0] Uses: v
convolution = g.map((_, i) => g.reduce((sum, z, j) => sum + z * padded[(i - j + 3) % 3], 0), ) Uses: g, padded

Returns

Example result

g: array (3 items): [1, 2, 3]
v: array (2 items): [1, 2]
padded: array (3 items): [1, 2, 0]
convolution: array (3 items): [7, 4, 7]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy uses d = 3 group positions and n = 2 window positions, so n < d.

Source: Source expression