The Modern Mathematics of Deep Learning

Linear regression with noise

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.

Observations and sampling · Describe examples, targets and samples.

The selected rows supply measurements and known answers to training; held-out IDs remain separate.

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

The target Y is the dot product of the true parameter θ* and input X plus a noise term ν.

The model separates the linear signal from noise.

Source formula example

Y=\langle\theta^{*},X\rangle+\nu

illustration · This fixes one realization; it does not specify the joint distribution of X and noise.

Symbols

$θ*$: The true parameter vector.
$X$: A random input vector.
$ν$: A noise term.
$Y$: The random target.

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

theta = [2, 1]
X = [3, 4]
noise = 0.5

Calculated here

Result expression: { theta, X, noise, signal, Y: signal + noise }

signal = theta.reduce((s, t, i) => s + t * X[i], 0) Uses: theta, X

Returns

Example result

theta: array (2 items): [2, 1]
X: array (2 items): [3, 4]
noise: 0.5
signal: 10
Y: 10.5

JavaScript

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

Result

Enable JavaScript to run this example.

Technical statement

The numerical values are invented. This fixes one realization; it does not specify the joint distribution of X and noise.

Source: Inline notation