#representation-geometry

Question 1: Single choice

An encoder $f$ maps data from $N$ groups to unit vectors; group $i$ lives on a finite set $S_i$ . Define $\mathrm{diameter}_i = \max_{s, t \in S_i}\lVert f(s) - f(t) \rVert$ and $\mathrm{margin}_{i \to j} = \min_{s \in S_i,\, t \in S_j}\lVert f(s) - f(t) \rVert$ . A contrastive "classification" draws an anchor and a positive from group $i$ and one negative from every other group, and is correct if the positive has the largest dot product with the anchor. Which condition guarantees that every such classification is correct?

For every group $i$ and every other group $j$ , $\mathrm{diameter}_i < \mathrm{margin}_{i \to j}$ 1
For every group $i$ there is some other group $j$ with $\mathrm{diameter}_i < \mathrm{margin}_{i \to j}$ 2
Some group $i^\star$ has a diameter smaller than every margin between groups3
The margins between all pairs of groups are equal4

#representation-geometry

Timed test