The Modern Mathematics of Deep Learning

A limiting kernel

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.

Kernels and linearization · Relate parameter gradients or features to kernel matrices.

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: 2.1 A view through similarity

K∞ is the deterministic kernel obtained in the stated infinite-width limit. A finite kernel example can show how a kernel compares two inputs.

The limiting kernel describes a simplified training regime.

Source formula example

K^{\infty}

illustration · The dot-product kernel illustrates a kernel; it is not derived as the limiting kernel for the source architecture.

Symbols

$K∞$: The limiting neural tangent kernel.
$x₁,x₂$: Input vectors.

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.

This example’s data flow

Initial values in code

x1 = [1, 2]
x2 = [3, 4]

Calculated here

Result expression: { x1, x2, kernel }

kernel = x1.reduce((s, v, i) => s + v * x2[i], 0) Uses: x1, x2

Returns

Example result

x1: array (2 items): [1, 2]
x2: array (2 items): [3, 4]
kernel: 11

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. The dot-product kernel illustrates a kernel; it is not derived as the limiting kernel for the source architecture.

Source: Inline notation