The Modern Mathematics of Deep Learning

Length of a feature trajectory

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: 3.3 Other ways to measure possibility

A curve in input space passes through the network. Its layer outputs trace another curve, whose length describes how far those outputs travel.

For γ(t)=t and feature ReLU(2t−1), t in [0,1], the output stays at zero for the first half and then rises to one.

Trajectory length is one measure used in the source discussion of expressivity.

Source formula example

\operatorname{Length}(\bar{\Phi}^{(\ell)}(\gamma(\cdot),\theta)),\quad\ell\in[L-1],

illustration · For this monotone one-dimensional example, the segment sum equals the exact trajectory length. A finite polygonal sample does not generally equal the length of an arbitrary curve.

Symbols

$γ$: A continuous input curve on [0,1].
$Φ̄^{(ℓ)}$: Activated output of layer ℓ.

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.

Positive scale

This example’s data flow

You choose

input.scale = 2

Initial values in code

points = [0, 0.5, 1]

Calculated here

Result expression: { points, output, segmentLengths, length: segmentLengths.reduce((a, b) => a + b, 0), }

output = points.map((t) => Math.max(0, input.scale * t - 1)) Uses: points, input.scale
segmentLengths = output.slice(1).map((value, i) => Math.abs(value - output[i])) Uses: output

Returns

Example result

points: array (3 items): [0, 0.5, 1]
output: array (3 items): [0, 0, 1]
segmentLengths: array (2 items): [0, 1]
length: 1

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy feature is monotone on [0,1], so its exact total variation equals its endpoint difference. The source studies trajectory length for random neural-network initializations and discusses how it changes with layer depth.

Source: Feature trajectory