The Modern Mathematics of Deep Learning

Dictionary projection

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: 7.2 A few building blocks at a time

Project x onto dictionary columns; if x = Dφ, the result is DᵀDφ.

The projection is the input to the thresholding map.

Source formula example

\psi\coloneqq D^{T}x=D^{T}D\phi

exact · This identity alone does not establish sparse recovery.

Symbols

$D$: Dictionary matrix.
$φ$: Coefficient vector.
$x$: Signal Dφ.
$ψ$: Projection Dᵀx.
$T$: 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 coefficient

This example’s data flow

You choose

input.first = 2

Initial values in code

D = [ [1, 0, 1], [1, 1, 0], [0, 1, 1], ]

Calculated here

Result expression: { phi, x, projection, gram }

phi = [input.first, 1, 0] Uses: input.first
x = multiply(D, phi) Uses: multiply, D, phi
projection = multiply(transpose(D), x) Uses: multiply, transpose, D, x
gram = multiply(transpose(D), multiply(D, phi)) Uses: multiply, transpose, D, phi

Returns

Example result

phi: array (3 items): [2, 1, 0]
x: array (3 items): [2, 3, 1]
projection: array (3 items): [5, 4, 3]
gram: array (3 items): [5, 4, 3]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The invented circulant dictionary represents a length-two filter on three circular positions, with n = 2 < d = 3. The code computes both sides using actual matrix products.

Source: Source expression