What Trace Powers Reveal About Log-Determinants: Closed-Form Estimators, Certificates, and Failure Modes

Abstract

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 $\log (\hat{I}»)$ around the arithmetic mean. This requires $|λ- \AM| < \AM$ and diverges when $\hat{I}º>4$. 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 $K^{\prime }(0)$. 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 $r\le \hat{I}{»}_{\min }$, 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 $m\in \{4,\dots ,8\}$, this is effectively constant time.