The Modern Mathematics of Deep Learning

Convolution and cross-correlation

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: 6.3 Taking apart and rebuilding

An inner product with a convolved filter equals an inner product with the corresponding cross-correlation.

This calculation explains the corresponding source expression.

Source formula example

\langle x,u_{i}\ast_{\circ}v_{j}\rangle

illustration · The numerical equality illustrates the finite cyclic identity; it does not prove the general framelet formula.

Symbols

$x$: Signal on a finite cyclic group.
$uᵢ$: Analysis vector.
$vⱼ$: Zero-padded window filter.
$∗\circ$: Circular convolution with padded filter.
$\star$: Cross-correlation.
$⟨·,·⟩$: Euclidean inner product.

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.first = 2

Initial values in code

x = [1, 2, 3]
u = [0, 1, 0]

Calculated here

Result expression: { x, u, v, conv, correlation, convolutionInner: dot(x, conv), correlationInner: dot(u, correlation), }

v = [input.first, 1, 0] Uses: input.first
conv = u.map((_, i) => u.reduce((s, z, j) => s + z * v[(i - j + 3) % 3], 0)) Uses: u, v
correlation = x.map((_, i) => v.reduce((s, z, j) => s + z * x[(i + j) % 3], 0)) Uses: x, v

Returns

Example result

x: array (3 items): [1, 2, 3]
u: array (3 items): [0, 1, 0]
v: array (3 items): [2, 1, 0]
conv: array (3 items): [0, 2, 1]
correlation: array (3 items): [4, 7, 7]
convolutionInner: 7
correlationInner: 7

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy uses three circular positions and a window of length 2 padded with one zero. It calculates both inner products and the full correlation vector.

Source: Source expression