The Modern Mathematics of Deep Learning

Shrinking and growing products

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.

Functions and derivatives · Describe maps and how their outputs change.

Project vectors have two features; the training matrix has n rows and two columns. Mathematical tools may operate on these values when their assumptions apply.

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: 6.2 Shortcuts between layers

Compare each factor magnitude with λ and compare the product derivative with the corresponding power of λ.

Repeated factors below one can shrink derivatives, while factors above one can grow them.

Source formula example

|x_{\ell}|<\lambda<1

illustration · These finite examples illustrate the conditions and derivative sizes, not a result for every network.

Symbols

$xₗ$: Real factor in position ℓ.
$λ$: Positive comparison threshold.
$π$: Product of factors.
$L$: Number of factors.

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.

First small factor

This example’s data flow

You choose

input.small = 0.5

Initial values in code

large = [2, 2, 2]
smallLambda = 0.75
largeLambda = 1.5

Calculated here

Result expression: { smallFactorsFit: small.every((v) => Math.abs(v) < smallLambda), smallDerivative: derivative(small), smallBound: smallLambda ** 2, largeFactorsFit: large.every((v) => Math.abs(v) > largeLambda), largeDerivative: derivative(lar...

small = [input.small, 0.5, 0.5] Uses: input.small

Returns

Example result

smallFactorsFit: true
smallDerivative: 0.25
smallBound: 0.5625
largeFactorsFit: true
largeDerivative: 4
largeBound: 2.25

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The toy has L = 3 and differentiates with respect to the first factor. Small factors are bounded by λ = 0.75 < 1; large factors exceed λ = 1.5 > 1.

Source: Source expression