The Modern Mathematics of Deep Learning

Downsampling a grid

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

Select every other row and column using one-based odd positions. JavaScript arrays use zero-based positions.

A pooling step can reduce the number of positions.

Source formula example

p:{\mathbb{R}}^{{G}}\to{\mathbb{R}}^{\widetilde{{{G}}}},\quad x\mapsto(x_{2i-1,2j-1})_{i,j=1}^{d}.

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

Symbols

$p$: The downsampling map.

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

grid = [ [1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16], ]

Calculated here

Result expression: grid .filter((_, i) => i % 2 === 0) .map((row) => row.filter((_, j) => j % 2 === 0))

Returns

Example result: array (2 items): [array (2 items): [1, 3], array (2 items): [9, 11]]

Some runtime paths or values are not resolved here.

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

This toy uses a 4-by-4 grid and produces a 2-by-2 grid. This particular operation selects values; it does not average them.

Source: Source expression