The Geometry of Conceptors in Neural-Activation Spaces

Conceptors represent concepts as graded collections of directions in neural-activation space rather than as single vectors. We develop a geometric account of how these representations depend on the contrast used to define a concept: opposing values in bipolar fitting, or presence and absence in unipolar fitting. We derive an exact decomposition of a centered pooled conceptor into an average pole filter, a mean-separation update, and a covariance-shape update. Experiments across five language models on sentiment, political leaning and stance, and depression expression show that both updates improve reconstruction of held-out contrasts. Additional directions preserve variation beyond the mean-difference line, and changing the comparator changes the benefit of these directions. Shape, size, and overlap each show up downstream: in amplification experiments, bipolar fits raise evidence for both poles more consistently than unipolar fits; conceptor size (the quota) correlates positively with layerwise probe performance; and cross-concept overlap quantifies interference under subspace removal, linking shared directions to Boolean composition. Together, these findings explain how fitting choices shape conceptor geometry and support bipolar fitting when the objective is to amplify both opposing poles.