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[2605.01192] Linear-Readout Floors and Threshold Recovery in Computation in Superposition

arxiv.org · 9,099 words · saved by 1 readers

Abstract:Two recent approaches to computation in superposition reach different recursive capacity regimes: Hänni et al. certify $\tilde{O}(d^{3/2})$ computable features in width $d$ via an approximate-linear recursive template, while Adler and Shavit reach near-quadratic capacity (up to logarithmic factors) using thresholded Boolean recovery. The main contribution of this paper is conceptual: we argue these results are not contradictory because they maintain different interface invariants, and we formalize the distinction. As a tool, we record a rank-trace Welch-type lower bound for biorthogonal linear readouts: for $F \gg d$, the worst-case off-diagonal cross-talk of any unit-diagonal linear readout is $\Omega(d^{-1/2})$, and the bound is tight on average for unit-norm tight frames. At quadratic feature load $F=d^2$, random-support threshold recovery succeeds for sparsities $s=O(d/\log d)$, while linear readouts still incur $\Omega(s/d)$ average per-coordinate squared error on Bernoulli sparse states. Matching the Welch floor against the published tolerance of the Hänni correction layer explains the $d^{3/2}$ scale as a compatibility threshold for that template, not a universal upper bound. Robust nonlinear reset beyond the Hänni template is left open.

Near-Floor Geometry Is Generic: Leverage Dispersion in Trained Overcomplete Codes Hector Borobia hecboar@doctor.upv.es VRAIN, Universitat Politècnica de València, Spain Elies Seguí-Mas esegui@upvnet.upv.es Universitat Politècnica de València,…

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