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About this paper
Hiroyasu Usami, Keisuke Hara, Ayato Tsuboi, Naohiko Matsuda
The authors show separate attention heads can require exponentially many atoms when replaced by one exact softmax readout. For W_d, a sum of d logistic sigmoids, d two-entry heads use 2d head–atom terms, while an equivalent single head needs 2^d atoms, even with redesigned keys, values, and positive masses. The lower bound holds for uniform error strictly below 1/2, not every model cache.
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