Spectral Indistinguishability and Learning-Curve Identifiability in Quadratic Networks
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Computer Science > Machine Learning arXiv:2610.20678 (cs) [Submitted on 14 Oct 2026] Title: Spectral Indistinguishability and Learning-Curve Identifiability in Quadratic Networks Authors: Dekun Yang.
计算机科学 > 机器学习 arXiv:2610.20678 (cs) [提交于 2026 年 10 月 14 日] 标题：二次网络中的谱不可区分性与学习曲线可识别性 作者：Dekun Yang。
Abstract: We study what spectral summaries and scalar learning curves reveal about centered quadratic networks. We construct analytic positive input densities with exactly Gaussian one- and two-coordinate marginals. Their complete fourth-moment operators are intertwined by an orthogonal map fixing the entire diagonal teacher algebra, so the corresponding anchored spectral measures agree.
摘要：我们研究了谱摘要和标量学习曲线对中心化二次网络的揭示作用。我们构建了具有精确高斯一维和二维坐标边际分布的解析正输入密度。它们的完整四阶矩算子通过一个固定整个对角教师代数的正交映射相互交织，因此相应的锚定谱测度是一致的。
For selected common teachers with a nonzero cycle invariant, their factorized loss curves differ because the map does not preserve matrix multiplication. Cycle length $r$, which can be arbitrarily large, gives $O(\varepsilon^r)$ contrast uniformly in training time for each fixed construction.
对于具有非零循环不变量的选定公共教师，它们的因子化损失曲线有所不同，因为该映射不保持矩阵乘法。循环长度 $r$ 可以任意大，对于每个固定的结构，它在训练时间内均匀地给出 $O(\varepsilon^r)$ 的对比度。
For a supplied pair, noisy scalar loss queries require order $\varepsilon^{-2r}$ observations, while bounded initialization interventions with scalar gradient readouts require order $\varepsilon^{-2}$. A shared-input score, equal to a paired-gradient difference, attains the optimal input-sample exponent without knowing the teacher.
对于给定的配对，噪声标量损失查询需要 $O(\varepsilon^{-2r})$ 阶的观测值，而带有标量梯度读出的有界初始化干预则需要 $O(\varepsilon^{-2})$ 阶。一种等于配对梯度差的共享输入评分，无需了解教师即可达到最优的输入样本指数。
An explicit family of diverging auxiliary teacher scales instead admits a nonzero finite limiting contrast at coupling $1/\log L$. Conversely, for Gaussian inputs and a rationally independent teacher spectrum, one exact scalar loss curve identifies the initial predictor up to sign conjugation.
相反，一个发散的辅助教师尺度显式族在耦合 $1/\log L$ 时允许非零的有限极限对比度。反之，对于高斯输入和有理独立的教师谱，一条精确的标量损失曲线可以将初始预测器识别到符号共轭的程度。
Within this rationally independent class, with nonzero diagonal inverse-state error in every coordinate and calibrated time and loss scales, it also identifies the unknown teacher spectrum and active dimension, modulo simultaneous rotations.
在此有理独立类中，若每个坐标都有非零的对角逆状态误差，且时间与损失尺度经过校准，它还能识别出未知的教师谱和有效维度（模同时旋转）。
For fixed dimension and distinct unordered teacher sums, dimension-minimal exact observations give generic joint local recovery. Finite exact global bounds hold for a known teacher and, on the active class, under a blind dimension cap. Exact certificates and full-weight replay verify the construction; numerical inverse checks reveal severe ill-conditioning.
对于固定维度和不同的无序教师和，维度最小的精确观测值可提供通用的联合局部恢复。对于已知教师，以及在有效类中且处于盲维度上限下，存在有限的精确全局界限。精确证书和全权重重放验证了该结构；数值逆检查揭示了严重的病态性。