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

arXiv:2601.12612v1 Announce Type: cross
Abstract: Computing $logdet(A)$ 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 $log(lambda)$ around the arithmetic mean. This requires $|lambda – AM| < AM$ and diverges when $kappa > 4$. We work instead with the moment-generating function $M(t) = E[X^t]$ for normalized eigenvalues $X = lambda/AM$. Since $M'(0) = E[log X]$, the log-determinant becomes $logdet(A) = n(log AM + M'(0))$ — 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 $r leq lambda_{min}$, we obtain moment-constrained lower bounds, yielding a provable interval for $logdet(A)$. 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.