The Modern Mathematics of Deep Learning

Convolution and support

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.

Approximation and depth · Study approximation error and network size under stated conditions.

The project supplies finite observations and fitted parameters. A held-out sample does not establish population bounds or prove a theorem.

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: 3.1 A rule based on distance

Support identifies where a function is nonzero. The source uses the Fourier convolution rule and bounds its support by adding a ball to each direction line.

The toy shows the same support-addition rule for circular convolution on five discrete positions.

Support restrictions help explain a shallow-network approximation obstruction.

Source formula example

\supp(\widehat{\Phi\varphi})=\supp(\hat{\Phi}*\hat{\varphi})\subset\bigcup_{j=1}^{n}\left(\spann\{w_{j}\}+B_{r}(0)\right).

illustration · The discrete cyclic example shows support inclusion, not the continuous Fourier-transform identity or the theorem. Cancellation can make the actual support smaller than the possible set.

Symbols

$\hatΦ$: Fourier transform of the shallow network.
$φ$: Square root of the measure density in the source construction.
$B_r(0)$: A radius-r ball around the origin.

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.

Second filter entry

This example’s data flow

You choose

input.second = 1

Initial values in code

a = [1, 0, 0, 2, 0]

Calculated here

Result expression: { convolution, supportA, supportB, possible, actual }

b = [1, input.second, 0, 0, 0] Uses: input.second
n = a.length Uses: a
convolution = a.map((_, i) => a.reduce((sum, value, j) => sum + value * b[(i - j + n) % n], 0), ) Uses: a, b, n
supportA = a.flatMap((value, i) => (value !== 0 ? [i] : [])) Uses: a
supportB = b.flatMap((value, i) => (value !== 0 ? [i] : [])) Uses: b
possible = [ ...new Set(supportA.flatMap((i) => supportB.map((j) => (i + j) % n))), ].sort((x, y) => x - y) Uses: supportA, supportB, n
actual = convolution.flatMap((value, i) => (value !== 0 ? [i] : [])) Uses: convolution

Returns

Example result

convolution: array (5 items): [1, 1, 0, 2, …]
supportA: array (2 items): [0, 3]
supportB: array (2 items): [0, 1]
possible: array (4 items): [0, 1, 3, 4]
actual: array (4 items): [0, 1, 3, 4]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source chooses the Fourier transform of φ as the indicator of a ball, and its shallow-network Fourier support lies on direction lines. Convolving enlarges those supports to a union of lines plus the ball. The finite cyclic analogue computes possible sums of support indices explicitly.

Source: Fourier support