Discussion Overview
The discussion revolves around the differentiability of the "l1"-norm, specifically addressing why the norm defined as ||x||_0 = |x_1| + |x_2| + ... + |x_N| is not differentiable. Participants explore geometric interpretations and seek clarification on the implications for matrix calculus.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants clarify that the correct formulation of the norm includes absolute values, stating that ||x||_0 = |x_1| + |x_2| + ... + |x_N| is necessary for it to be a norm.
- One participant mentions that the non-differentiability arises because the unit sphere defined by this norm has "corners," drawing a parallel to the function f(x) = |x|, which is not differentiable at x=0.
- Another participant expresses appreciation for the geometric explanation but requests further elaboration on the topic in the context of matrix calculus, indicating a need for deeper understanding.
- There is a reiteration that the non-differentiability at extreme points of the unit sphere, such as (1,0,...,0), contributes to the overall confusion regarding the norm's properties.
Areas of Agreement / Disagreement
Participants generally agree on the necessity of absolute values in defining the norm and recognize the geometric reasoning behind its non-differentiability. However, there remains some confusion regarding its implications in matrix calculus, indicating that the discussion is not fully resolved.
Contextual Notes
Some assumptions about the definitions of norms and differentiability are present, and the discussion highlights the need for clarity in mathematical contexts, particularly regarding matrix calculus.