Smooth functions
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.
Functions and derivatives · Describe maps and how their outputs change.
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: 3.2 Simple pieces, many layers · 4.1 When data live on a smaller shape
A Cᵏ function has continuous derivatives through order k on the stated domain. A quadratic is C².
Smoothness assumptions control which functions a theorem covers.
Source formula example
illustration · The quadratic is a simple smooth example; one point evaluation cannot prove regularity for an arbitrary function.
Symbols
- A function on the indicated domain or manifold.
- A nonnegative integer derivative order.
- Functions with continuous derivatives through order k.
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
Code changed. Reset to restore this example’s data flow.
Initial values in code
x=2
Calculated here
Result expression: { x, value: x * x, firstDerivative: 2 * x, secondDerivative: 2 }
Returns
x- 2
value- 4
firstDerivative- 4
secondDerivative- 2
Read numbers from input and return the result. JavaScript basics.
Result
Technical statement
The numerical values are invented. The quadratic is a simple smooth example; one point evaluation cannot prove regularity for an arbitrary function.
Source: Inline notation