如何用文字比较不变性程度？规范表述函数不变性差异的方法

Great question—talking about degrees of invariance can feel ambiguous without standard ways to phrase it, so let’s break this down into your two core questions.

1. Comparing Degrees of Invariance in Plain Language #

When discussing how "invariant" one function/quantity is relative to another, you’ll want to tie the comparison to specific transformations or transformation classes. Here are intuitive, precise ways to phrase it:

• If one function stays invariant under a wider range of transformations:
  • "Function f has greater invariance than g with respect to [transformation type]" (e.g., "f has greater scale invariance than g")
  • "f is invariant under a larger class of transformations than g" (e.g., "f is invariant under all rotations between 0° and 90°, while g only stays invariant under 0° rotations")
• If a function doesn’t stay perfectly invariant but changes less drastically under transformation:
  • "f exhibits near-invariance to [transformation] to a greater degree than g"
  • "f is more robust to [transformation] than g" (this is especially common in fields like machine learning and signal processing)

2. Formalizing "One Function Changes Less Than Another Under a Transformation" #

The phrase "f is more scale invariant than g" is acceptable in informal contexts, but for technical precision, here are better alternatives depending on your use case:

Qualitative (Non-Quantified) #

• "f has lower sensitivity to [specific transformation] than g" (e.g., "f has lower sensitivity to scaling than g")
• "f demonstrates greater invariance robustness to [transformation] than g"
• "Under [transformation], f is more nearly invariant than g"

Quantitative (Precise, Measurement-Based) #

If you can quantify the change (e.g., using a norm, error metric, or percentage difference), you can be explicit:

• "For all x in the domain, the magnitude of change in f under [transformation T] is smaller than that in g: ||f(T(x)) - f(x)|| < ||g(T(x)) - g(x)||" (where ||·|| denotes a chosen norm, like L2 or pixel intensity difference)
• "When applying [transformation parameter k], the relative change in f is less than that in g: |f(k·x)/f(x) - 1| < |g(k·x)/g(x) - 1| for all x and valid k"

Example for Scaling Transformations #

Suppose scaling an image by 1.5 causes f’s output to shift by 5% (pixel intensity difference) and g’s output to shift by 20%. You could say:

"Under scaling transformations, f has lower sensitivity than g—its output changes only 5% versus g’s 20% shift. This means f is more nearly scale-invariant than g."

内容的提问来源于stack exchange，提问作者Faur