Polynomial-time SDP and ellipsoid-based approximation of Kolmogorov widths yields efficient robust detection boundaries matching upper bounds up to polylog factors for structured constrained signals.
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ABGD parametrizes piecewise linear functions as difference of max-affine functions and converges linearly to an epsilon-accurate solution with O(d max(sigma/epsilon,1)^2) samples under sub-Gaussian noise, which is minimax optimal up to logs.
The paper proves statistical consistency of contrastive loss to optimal ranking via an AUC criterion and derives generalization bounds O(1/m + 1/sqrt(n)) for supervised and O(1/sqrt(m) + 1/sqrt(n)) for self-supervised CRL that explain benefits of large negative sets.
Pre-training provides a geometric warm start in a single-index model that enables weak-to-strong generalization up to a supervisor-limited bound, with empirical phase-transition evidence in LLMs.
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