The Modern Mathematics of Deep Learning

A matrix near the identity

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.

Vectors and matrices · Combine numbers using vector and matrix operations.

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: 1.2 Learning from examples

Compare the operator norm with the number of columns times the largest absolute entry.

This example makes the source statement computable on a finite model.

Source formula example

\|\widetilde{W}^{(2)}\widetilde{A}-\mathrm{I}_{m}\|_{\mathrm{op}}\leq m\max_{i,j\in[m]}\big|(\widetilde{W}^{(2)}\widetilde{A}-\mathrm{I}_{m})_{i,j}\big|=m\max_{j\in[m]}\|\Phi_{\tilde{a}}(x^{(j)},\tilde{\theta})-g(x^{(j)})\|_{\infty}<1,

illustration · This checks the entire inequality chain for a diagonal toy; it does not construct the source approximation network.

Symbols

$W̃⁽²⁾$: Chosen output-weight matrix.
$Ã$: Hidden activation matrix.
$Iₘ$: m by m identity.
$\Vert ·\Vert op$: Euclidean operator norm.
$\Vert ·\Vert ∞$: Largest absolute vector entry.
$m$: Number of training points.

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.

Diagonal error

This example’s data flow

You choose

input.epsilon = 0.1

Calculated here

Result expression: { residual, operatorNorm, maxEntry, maxColumnInfinityNorm, bound: 2 * maxEntry, belowOne: 2 * maxEntry < 1, }

epsilon = input.epsilon Uses: input.epsilon
residual = [ [epsilon, 0], [0, -epsilon], ] Uses: epsilon
operatorNorm = Math.abs(epsilon) Uses: epsilon
maxEntry = Math.abs(epsilon) Uses: epsilon
maxColumnInfinityNorm = Math.abs(epsilon) Uses: epsilon

Returns

Example result

residual: array (2 items): [array (2 items): [0.1, 0], array (2 items): [0, -0.1]]
operatorNorm: 0.1
maxEntry: 0.1
maxColumnInfinityNorm: 0.1
bound: 0.2
belowOne: true

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

Take m=2, W̃⁽²⁾=I and Ã=diag(1+ε,1−ε); target columns are the identity columns. The diagonal residual has operator norm |ε|, and the largest column error equals the maximum absolute entry.

Source: Source expression