The Modern Mathematics of Deep Learning

Convolution then downsampling

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.

Convolution · Describe shared filters and downsampling.

Convolution, recurrent and residual architectures are not part of this linear penguin model.

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: 6.1 Looking locally

Each channel uses its own shared filter, then the same downsampling rule selects positions.

These operations define structured network layers and their analysis.

Source formula example

{\mathbb{R}}^{{G}}\ni x\mapsto x^{\prime}\coloneqq(p(x*\kappa_{i}))_{i=1}^{C}\in({\mathbb{R}}^{\widetilde{{{G}}}})^{C}.

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

Symbols

$G$: Finite group of input positions.
$G̃$: Output positions after downsampling.
$κᵢ$: Filter for output channel i.
$p$: Downsampling map.
$C$: Positive number of output channels.

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.

This example’s data flow

Initial values in code

x = [1, 2, 3, 4]
filters = [ [1, 0, 0, 0], [0, 1, 0, 0], ]

Calculated here

Result expression: { channels, downsampled: channels.map((c) => c.filter((_, i) => i % 2 === 0)) }

channels = filters.map((f) => x.map((_, i) => f.reduce((sum, v, j) => sum + v * x[(i - j + 4) % 4], 0)), ) Uses: filters, x

Returns

Example result

channels: array (2 items): [array (4 items): [1, 2, 3, 4], array (4 items): [4, 1, 2, 3]]
downsampled: array (2 items): [array (2 items): [1, 3], array (2 items): [4, 2]]

JavaScript

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

Result

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Technical statement

The numbers in this example are invented.

Source: Source expression