The Modern Mathematics of Deep Learning

A constant-kernel flow solution

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

The source writes the prediction along a constant-kernel gradient-flow trajectory using a matrix expression C(t).

For one training point with kernel value 1, target 2, initial prediction 0, and squared loss, C(t) = 1 − exp(−2t).

This special case shows predictions approaching the target over time.

Source formula example

\Phi({\cdot},\Theta(t))=C(t)(y^{(i)})_{i=1}^{m}+\big(\Phi(\cdot,\Theta_{0})-C(t)(\Phi(x^{(i)},\Theta_{0}))_{i=1}^{m}\big),

illustration · The code solves only the stated one-point constant-kernel case. It does not establish that an arbitrary finite network has a constant kernel.

Symbols

$C(t)$: The source matrix-valued coefficient, reduced to one scalar in the toy.
$Θ_0$: Initial parameters.

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.

Time t

This example’s data flow

You choose

input.t = 1

Initial values in code

target = 2
initialPrediction = 0

Calculated here

Result expression: { c, target, initialPrediction, prediction }

c = 1 - Math.exp(-2 * input.t) Uses: input.t
prediction = c * target + (initialPrediction - c * initialPrediction) Uses: c, target, initialPrediction

Returns

Example result

c: ≈0.8647
target: 2
initialPrediction: 0
prediction: ≈1.729

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The source uses squared loss and a limiting kernel whose sample kernel matrix is invertible. C(t) contains that kernel matrix, its inverse, and a matrix exponential. With one point, kernel 1, and m=1, the scalar expression is 1−exp(−2t). The complete source expression includes the initial prediction at the query and training points.

Source: Source expression