The Modern Mathematics of Deep Learning

Window and circular 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

A window dot product can also be computed as an inner product with the circularly shifted filter.

This equality links the window matrix to convolution.

Source formula example

\langle e^{(i)},\mathbb{H}_{n}(x)v\rangle=\sum_{j\in H}x_{i+j}v_{j}=\langle x,e^{(i)}\ast_{\circ}v\rangle,\quad i\in G,

exact · A finite cyclic example illustrates the general group identity.

Symbols

$G$: Finite cyclic group of positions.
$H$: Window index set.
$e⁽ⁱ⁾$: Basis vector selecting position i.
$v$: Window vector.
$Hₙ(x)$: Circular window matrix.
$∗\circ$: Circular convolution.

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.

Position

This example’s data flow

You choose

input.position = 1

Initial values in code

x = [1, 2, 3]
v = [2, 1]
padded = [2, 1, 0]

Calculated here

Result expression: { window, shifted, windowDot: dot(window, v), correlationDot: dot(x, shifted) }

i = input.position Uses: input.position
window = v.map((_, j) => x[(i + j) % 3]) Uses: v, x, i
shifted = x.map((_, k) => padded[(k - i + 3) % 3]) Uses: x, padded, i

Returns

Example result

window: array (2 items): [2, 3]
shifted: array (3 items): [0, 2, 1]
windowDot: 7
correlationDot: 7

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy has a three-position circular group and a length-two window, satisfying n < d. The window vector is zero-padded before circular convolution.

Source: Source expression