The Modern Mathematics of Deep Learning

Approximation size scales

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 · 4.1 When data live on a smaller shape

The source discusses these expressions as size scales under different structural assumptions. Big-O notation hides constants and describes behavior as ε becomes small.

The first two scales arise in the radial-function discussion; the third occurs for localized functions in the manifold argument.

The required size can depend strongly on dimension and smoothness.

Source formula example

{\mathcal{O}}(C/{\varepsilon})

illustration · The code evaluates visible factors. Big-O constants, precise network sizes and approximation errors are not computed.

Symbols

$C$: A Lipschitz constant in the first scale.
$ε$: Positive target error.
$d$: Ambient dimension.
$d^{\prime}$: Intrinsic manifold dimension.
$k$: Smoothness order.

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.

Lipschitz constant CAccuracy εDimension dIntrinsic dimension d′Smoothness k

This example’s data flow

You choose

input.C = 2
input.epsilon = 0.5
input.d = 2
input.intrinsic = 1
input.k = 2

Calculated here

Result expression: { oneDimensionalScale: C / epsilon, squaredNormScale: d ** 2 / epsilon, manifoldScale: epsilon ** (-(intrinsic + 1) / k), }

C = input.C Uses: input.C
epsilon = input.epsilon Uses: input.epsilon
d = input.d Uses: input.d
intrinsic = input.intrinsic Uses: input.intrinsic
k = input.k Uses: input.k

Returns

Example result

oneDimensionalScale: 4
squaredNormScale: 8
manifoldScale: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The radial discussion uses a C-Lipschitz univariate function on [0,1] and linear interpolation, then discusses squared Euclidean norm on [0,1]ᵈ. The manifold discussion uses a compact embedded manifold, local charts and sufficiently smooth localized functions; its localized approximation scale is stated as ε→0. Each big-O expression hides constants and has its own assumptions.

Source: Approximation size scales