The Modern Mathematics of Deep Learning

Circular 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: 6.1 Looking locally

On a circular list, indices wrap around. Multiply matching shifted entries and add.

Convolution shares a filter across positions.

Source formula example

(a*b)_{i}=\sum_{j\in{{G}}}a_{j}b_{j^{-1}i},\quad i\in{{G}}.

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

Symbols

$G$: Here the finite cyclic group of three indices.
$j^{-1}i$: Group operation: i − j modulo 3 in this example.

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 filter value

This example’s data flow

You choose

input.a = 1

Initial values in code

b = [1, 0, 1]

Calculated here

Result expression: { a, b, output }

a = [input.a, 2, 3] Uses: input.a
output = a.map((_, i) => a.reduce((sum, value, j) => sum + value * b[(i - j + a.length) % a.length], 0), ) Uses: a, b

Returns

Example result

a: array (3 items): [1, 2, 3]
b: array (3 items): [1, 0, 1]
output: array (3 items): [3, 5, 4]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source gives convolution on a group. This example uses the three-element cyclic group; its indexing wraps at both ends.

Source: Source expression