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

TL;DR

This paper introduces new trace power-based methods for estimating the log-determinant of large matrices, providing closed-form estimators, bounds, and failure mode analysis with guarantees and efficiency.

Contribution

It develops a novel approach using the moment-generating function transform for spectral estimation, establishing fundamental limits and practical bounds for log-determinant approximation.

Findings

Proves no continuous estimator can be uniformly accurate over unbounded spectra.

Derives upper bounds on the geometric mean of eigenvalues from trace moments.

Provides spectral gap diagnostics to assess estimator reliability.

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 $\log(\lambda)$ 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 $\log\det(A) = n(\log \AM + M'(0))$ -- 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 $r \leq \lambda_{\min}$, 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.