The Modern Mathematics of Deep Learning

A one-sample matrix pseudoinverse

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: 2.4 A second descent

With one sample whose d coordinates are ±1, its squared norm is d. The source gives this explicit pseudoinverse of its outer-product matrix.

The toy sample is X=(1,−1), so d=2.

The pseudoinverse appears in the source variance calculation.

Source formula example

\Sigma^{+}=\frac{X^{(1)}(X^{(1)})^{T}}{\|X^{(1)}\|^{4}}=\frac{X^{(1)}(X^{(1)})^{T}}{d^{2}}

exact · The result is exact for these finite toy inputs, apart from ordinary computer rounding.

Symbols

$Σ^+$: Moore–Penrose pseudoinverse of Σ.
$X^{(1)}$: One binary vector in {−1,1}ᵈ.

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.

Second coordinate

This example’s data flow

You choose

input.second = -1

Calculated here

Result expression: { x, normSquared, outer, pseudoinverse, projection }

x = [1, input.second] Uses: input.second
normSquared = x.reduce((sum, value) => sum + value * value, 0) Uses: x
outer = x.map((a) => x.map((b) => a * b)) Uses: x
pseudoinverse = outer.map((row) => row.map((value) => value / normSquared ** 2)) Uses: outer, normSquared
projection = outer.map((row, i) => row.map( (_, j) => (i === j ? 1 : 0) - pseudoinverse[i].reduce((sum, value, k) => sum + value * outer[k][j], 0), ), ) Uses: outer, pseudoinverse

Returns

Example result

x: array (2 items): [1, -1]
normSquared: 2
outer: array (2 items): [array (2 items): [1, -1], array (2 items): [-1, 1]]
pseudoinverse: array (2 items): [array (2 items): [0.25, -0.25], array (2 items): [-0.25, 0.25]]
projection: array (2 items): [array (2 items): [0.5, 0.5], array (2 items): [0.5, 0.5]]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

For a nonzero vector x, the pseudoinverse of xxᵀ is xxᵀ/‖x‖⁴. Binary coordinates make ‖x‖²=d, giving the second expression. This code uses d=2 and also computes P=I−Σ⁺Σ, the orthogonal projection onto the kernel of Σ.

Source: One-sample pseudoinverse