The Modern Mathematics of Deep Learning

Batch normalization Jacobian

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.

Batch normalization · Describe batch normalization and its derivatives.

The project uses a linear rule with two features. The fitted weights produce scores or normalized mass.

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: 6.4 Adjusting the scale

The Jacobian records how every normalized batch value changes when each batch input changes.

These operations define structured network layers and their analysis.

Source formula example

\frac{\partial\mathcal{B}^{(\beta,\gamma)}({b})}{\partial{b}}=\frac{\gamma}{\sigma_{b}}\Big(\mathrm{I}_{m}-\frac{1}{m}\mathbf{1}\mathbf{1}^{T}-\frac{1}{m}\mathcal{B}^{(0,1)}({b})(\mathcal{B}^{(0,1)}({b}))^{T}\Big),\quad{b}\in{\mathbb{R}}^{m},

illustration · The invented finite inputs illustrate this expression. They do not prove a general identity, bound, or existence claim.

Symbols

$b$: A real batch vector of length m.
$γ$: Real scale.
$σb$: Positive batch standard deviation.
$B⁽⁰,¹⁾$: Batch normalized with zero shift and unit scale.
$Iₘ$: m by m identity matrix.
$1$: Length-m vector of ones.

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.

Scale

This example’s data flow

You choose

input.gamma = 1

Initial values in code

b = [-1, 0, 1]

Calculated here

Result expression: { jacobian, rowSums: jacobian.map((row) => row.reduce((a, v) => a + v, 0)) }

sigma = Math.sqrt(2 / 3)
z = b.map((v) => v / sigma) Uses: b, sigma
jacobian = b.map((_, i) => b.map( (__, j) => (input.gamma / sigma) * ((i === j ? 1 : 0) - 1 / 3 - (z[i] * z[j]) / 3), ), ) Uses: b, input.gamma, sigma, z

Returns

Example result

jacobian: array (3 items): [array (3 items): [≈0.2041, ≈-0.4082, ≈0.2041], array (3 items): [≈-0.4082, ≈0.8165, ≈-0.4082], array (3 items): [≈0.2041, ≈-0.4082, ≈0.2041]]
rowSums: array (3 items): [0, 0, 0]

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The batch is scalar-valued, has three entries and nonzero standard deviation. This computes the stated Jacobian at one batch; it does not prove the gradient norm identity. the source proposition takes m ≥ 2, real β and γ, and differentiable r. The formula requires σb > 0 even though the displayed source statement says every batch; constant batches make its denominator zero.

Source: Source expression