The Axiotis-Sviridenko Condition-Number Conjecture
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Statement
Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. Their conjectured lower bound is established for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer and Tulsiani.
Context
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
A 2021 conjecture on the price of sparsity in least squares, sitting in the hardness-of-approximation literature that grew from the Small-Set Expansion Hypothesis.
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The acknowledgements state that the proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google, with the authors verifying it and editing for presentation.
Conditional on the randomized exact-volume Small-Set Expansion Hypothesis, and stated for least-squares objectives rather than sparse convex optimization in general.
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