The Modern Mathematics of Deep Learning

Dual-frame matrix identity

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

When analysis and synthesis matrices multiply to the identity, inserting them on both sides leaves a matrix unchanged.

These operations define structured network layers and their analysis.

Source formula example

\mathbb{H}_{n}(x)=\widetilde{U}U^{T}\mathbb{H}_{n}(x)V\widetilde{V}^{T}.

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

Symbols

$Hₙ(x)$: Circular window matrix.
$U,V$: Matrices of analysis frame vectors.
$\tilde{U},Ṽ$: Matrices of dual synthesis frame vectors.
$T$: Matrix transpose.

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 entry

This example’s data flow

You choose

input.x = 1

Initial values in code

leftIdentity = [ [1, 0, 0], [0, 1, 0], [0, 0, 1], ]
rightIdentity = [ [1, 0], [0, 1], ]
p = 3
q = 2
d = 3
n = 2

Calculated here

Result expression: { leftIdentity, rightIdentity, p, q, d, n, original: H, reconstructed: multiply(multiply(leftIdentity, H), rightIdentity), }

H = [ [input.x, 2], [2, 3], [3, input.x], ] Uses: input.x

Returns

Example result

original: array (3 items): [array (2 items): [1, 2], array (2 items): [2, 3], array (2 items): [3, 1]]
reconstructed: array (3 items): [array (2 items): [1, 2], array (2 items): [2, 3], array (2 items): [3, 1]]
leftIdentity: array (3 items): [array (3 items): [1, 0, 0], array (3 items): [0, 1, 0], array (3 items): [0, 0, 1]]
rightIdentity: array (2 items): [array (2 items): [1, 0], array (2 items): [0, 1]]
p: 3
q: 2
d: 3
n: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The identity matrices are special pairs of dual frames. Their dimensions are d = 3 and n = 2, with n < d. The source requires the stated dual-frame identities for both dimensions. The displayed H is the circular window matrix of x = [input.x, 2, 3]. In this basis frame p=d=3 and q=n=2, so p≥d and q≥n. The columns of the shown identity matrices are the vectors uᵢ and vⱼ.

Source: Source expression