The Modern Mathematics of Deep Learning

Matrix operator norm

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.

Norms and distances · Measure size or difference in a specified space.

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: 1.1 A note on symbols

The operator norm is the largest ratio of output length to input length over nonzero input vectors.

It measures the largest possible scaling by a matrix.

Source formula example

\|\cdot\|_{\mathrm{op}}

illustration · The maximum absolute diagonal entry gives the Euclidean operator norm for this diagonal matrix; a general matrix needs a different calculation.

Symbols

$A$: A real matrix.
$\Vert A\Vert op$: Its Euclidean induced operator norm.

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.

Value

This example’s data flow

You choose

input.x = 2

Calculated here

Result expression: { diagonal, operatorNorm: Math.max(Math.abs(a), 1) }

a = input.x Uses: input.x
diagonal = [a, 1] Uses: a

Returns

Example result

diagonal: array (2 items): [2, 1]
operatorNorm: 2

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values in this example are invented. The maximum absolute diagonal entry gives the Euclidean operator norm for this diagonal matrix; a general matrix needs a different calculation.

Source: Inline notation