The Modern Mathematics of Deep Learning

A low-rank window matrix

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 rank-r window matrix has r nonzero singular directions.

This example makes the source notation or condition concrete.

Source formula example

p=q=r\leq n

illustration · This rank-one example does not characterize arbitrary window matrices.

Symbols

$Hₙ(x)$: d by n circular window matrix.
$r$: Matrix rank.
$p,q$: Numbers of selected left and right singular vectors.
$n$: Window size.

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.

Nonzero entry

This example’s data flow

You choose

input.value = 1

Initial values in code

d = 3
n = 2
r = 1
p = 1
q = 1

Calculated here

Result expression: { H, d, n, p, q, r, singularValue, rankCondition: p === q && q === r && r <= n }

value = input.value Uses: input.value
H = [ [value, value], [value, value], [value, value], ] Uses: value
singularValue = Math.sqrt(6) * Math.abs(value) Uses: value

Returns

Example result

H: array (3 items): [array (2 items): [1, 1], array (2 items): [1, 1], array (2 items): [1, 1]]
d: 3
n: 2
p: 1
q: 1
r: 1
singularValue: ≈2.449
rankCondition: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

For x=[value,value,value] with value>0, H is a nonzero 3 by 2 constant matrix of rank 1, and its only nonzero singular value is √6 times value. The source chooses p=q=r≤n in its singular-vector discussion.

Source: Source expression