The Modern Mathematics of Deep Learning

Minimum kernel norm interpolation

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

Find the smallest norm among functions that fit every training point.

The source identifies a minimum-norm interpolant in an infinite-width kernel regime.

Source formula example

\min_{f\in\mathcal{H}_{K^{\infty}}}\|f\|_{K^{\infty}}\quad\text{s.t.}\quad f(x^{(i)})=y^{(i)}.

illustration · This toy does not identify the limiting neural tangent kernel or prove convergence to its interpolant.

Symbols

$Hᴷ∞$: Reproducing-kernel Hilbert space associated with K∞.
$\Vert f\Vert K∞$: Kernel-space norm.
$x⁽ⁱ⁾,y⁽ⁱ⁾$: Training inputs and targets.

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.

Target

This example’s data flow

You choose

input.target = 2

Initial values in code

x = 1

Calculated here

Result expression: { weight, prediction: weight * x, norm: Math.abs(weight), minimumNorm: Math.abs(weight), }

y = input.target Uses: input.target
weight = y / x Uses: y, x

Returns

Example result

weight: 2
prediction: 2
norm: 2
minimumNorm: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Use the invented linear kernel K(x,z)=xz on real inputs. Its functions are f(x)=wx with norm |w|. The single constraint f(1)=target fixes w uniquely, so this is the exact minimum.

Source: Source expression