用矩阵幂迹估计对数行列式，实现快速高精度计算与可信度诊断。
What Trace Powers Reveal About Log-Determinants: Closed-Form Estimators, Certificates, and Failure Modes
- 通过归一化特征值的矩生成函数，将对数行列式转化为导数估计问题。
- 在4到8个迹值下，计算复杂度恒定，且能提供保证的上下界。
- 内置故障诊断机制，可判断点估计是否可靠，适合科学计算场景。
大规模对称正定矩阵的logdet(A)计算常见于高斯过程推断和贝叶斯模型比较。传统方法依赖矩阵-向量乘积与多项式逼近。本文研究另一种模式：当矩阵幂可访问时，可获取迹幂p_k = tr(A^k)。经典基于矩的方法对特征值均值作泰勒展开，要求|λ - AM| < AM，当条件数κ > 4时发散。本文转而使用归一化特征值X = λ/AM的矩生成函数M(t) = E[X^t]，因M'(0) = E[log X]，故logdet(A) = n(log AM + M'(0))。迹幂提供M(k)在正整数上的取值，但直接插值M(t)因指数增长而病态。引入变换K(t) = log M(t)，归一化使K(0) = K(1) = 0，固定两端后通过m+1个连续整数插值K(t)，求导得K'(0)。然而局部插值无法捕捉任意谱特性。本文证明根本限制：仅用有限正矩的连续估计器无法在无界条件下均匀精确。正矩会弱化谱尾部影响；而K'(0) = E[log X]对尾部敏感。因此提出保证性上界，利用迹幂得到(det A)^{1/n}的上界，并在已知最小特征值下界r ≤ λ_min时，构造矩约束下的下界，形成对logdet(A)的可信区间。间隙诊断指标可判断点估计是否可信。所有估计与边界计算代价为O(m)，与n无关。当m ∈ {4, ..., 8}时，等价常数时间。
原文摘要 · Abstract (English)
Computing $\log\det(A)$ for large symmetric positive definite matrices arises in Gaussian process inference and Bayesian model comparison. Standard methods combine matrix-vector products with polynomial approximations. We study a different model: access to trace powers $p_k = \tr(A^k)$, natural when matrix powers are available. Classical moment-based approximations Taylor-expand $\log(λ)$ around the arithmetic mean. This requires $|λ- \AM| < \AM$ and diverges when $κ> 4$. We work instead with the moment-generating function $M(t) = \E[X^t]$ for normalized eigenvalues $X = λ/\AM$. Since $M'(0) = \E[\log X]$, the log-determinant becomes $\log\det(A) = n(\log \AM + M'(0))$ -- the problem reduces to estimating a derivative at $t = 0$. Trace powers give $M(k)$ at positive integers, but interpolating $M(t)$ directly is ill-conditioned due to exponential growth. The transform $K(t) = \log M(t)$ compresses this range. Normalization by $\AM$ ensures $K(0) = K(1) = 0$. With these anchors fixed, we interpolate $K$ through $m+1$ consecutive integers and differentiate to estimate $K'(0)$. However, this local interpolation cannot capture arbitrary spectral features. We prove a fundamental limit: no continuous estimator using finitely many positive moments can be uniformly accurate over unbounded conditioning. Positive moments downweight the spectral tail; $K'(0) = \E[\log X]$ is tail-sensitive. This motivates guaranteed bounds. From the same traces we derive upper bounds on $(\det A)^{1/n}$. Given a spectral floor $r \leq λ_{\min}$, we obtain moment-constrained lower bounds, yielding a provable interval for $\log\det(A)$. A gap diagnostic indicates when to trust the point estimate and when to report bounds. All estimators and bounds cost $O(m)$, independent of $n$. For $m \in \{4, \ldots, 8\}$, this is effectively constant time.
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