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.

翻译：对于大规模对称正定矩阵计算 $\log\det(A)$ 的问题，常见于高斯过程推断与贝叶斯模型比较。标准方法将矩阵-向量乘积与多项式逼近相结合。本文研究一种不同的模型：通过迹幂 $p_k = \tr(A^k)$ 进行估计，这在矩阵幂可高效计算时尤为自然。经典的基于矩的逼近方法将 $\log(λ)$ 在算术均值 $\AM$ 处进行泰勒展开。该方法要求 $|λ- \AM| < \AM$，且在条件数 $κ> 4$ 时发散。我们转而利用归一化特征值 $X = λ/\AM$ 的矩生成函数 $M(t) = \E[X^t]$。由于 $M'(0) = \E[\log X]$，对数行列式可表示为 $\log\det(A) = n(\log \AM + M'(0))$——问题转化为在 $t = 0$ 处估计导数。迹幂提供了 $M(k)$ 在正整数处的值，但由于指数增长特性，直接插值 $M(t)$ 是病态的。变换 $K(t) = \log M(t)$ 压缩了其值域。通过 $\AM$ 归一化确保 $K(0) = K(1) = 0$。固定这两个锚点后，我们通过 $m+1$ 个连续整数点对 $K$ 进行插值，并通过微分估计 $K'(0)$。然而，这种局部插值无法捕捉任意的谱特征。我们证明了一个基本极限：对于无界条件数，任何使用有限正矩的连续估计器均无法保证一致精度。正矩会削弱谱尾的影响，而 $K'(0) = \E[\log X]$ 对谱尾敏感。这促使我们寻求可证明的界。基于相同的迹信息，我们推导出 $(\det A)^{1/n}$ 的上界。给定谱下界 $r \leq λ_{\min}$，我们得到矩约束下的下界，从而为 $\log\det(A)$ 提供一个可证明的区间。通过间隙诊断指标，可判断何时信任点估计、何时应报告区间界。所有估计器与界的计算成本均为 $O(m)$，与 $n$ 无关。当 $m \in \{4, \ldots, 8\}$ 时，这相当于常数时间计算。