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

TL;DR

The paper revisits log-determinant estimation for SPD matrices under a data model that only provides trace powers p_k, revealing an information gap that makes traditional eigenvalue Taylor expansions ill-suited when κ is large. It introduces an exponent-space approach using M(t)=E[X^t] and K(t)=log M(t), deriving the k0:m family of estimators via degree-m Lagrange interpolation to recover K'(0) and thus log det(A) with cost O(m), independent of n. The authors prove fundamental impossibility results that no finite-moment estimator can be universally accurate across all spectra, and they propose robust, computable guaranteed bounds based on k+1-atom optimizations that certify intervals for log det and GM/AM, with a practical gap diagnostic. They also analyze the impact of noisy trace inputs, provide guidance for adaptive order selection (favoring m≈4 under typical Hutchinson noise), and validate the framework across multiple spectrum types, demonstrating when point estimates suffice and when bounds are essential. Overall, the work delivers a principled, scalable framework for certified log-determinant estimation from trace powers, with concrete deployment rules and insightful diagnostics grounded in spectral structure.

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 $\log(λ)$ 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.

Figures (12)

• Figure 1: Absolute error in estimating $K'(0) = \log(\operatorname{GM}/\operatorname{AM})$, where $\operatorname{GM} = (\det A)^{1/n}$ is the geometric mean and $\operatorname{AM} = \operatorname{tr}(A)/n$ the arithmetic mean, across three spectrum types ($n=1024$, condition number $\kappa=100$). Latané's $\lambda$-space Taylor expansion (red) diverges exponentially with order. Our exponent-space interpolation (blue) achieves minimum error near $m \in \{4,5,6\}$ and remains stable at higher orders. For the geometric spectrum, our method achieves error $\approx 0.02$ at $m=5$ while Latané's error exceeds $10^3$ at $m=12$.
• Figure 2: Scaling with $n$ for lognormal eigenvalues ($\sigma = 0.5$). Left: bias $\approx 0$. Middle/Right: SD and RMSE decay as $O(n^{-0.5})$.
• Figure 3: Best achievable error in log-geometric-mean estimation. For each condition number $\kappa$, we compute the $k_{0:m}$ estimator for $m = 2, \ldots, 20$ and report the minimum absolute error $|\hat{K}'(0) - K'(0)|$ over all $m$. Two-point spectra (red) have the smallest Taylor radius $R = \pi/\log\kappa$ and exhibit the highest error floor; log-uniform spectra (blue) have $R = 2\pi/\log\kappa$; uniform spectra (cyan) have the largest $R$ and lowest floor. Vertical dotted lines mark $\kappa$ thresholds where $R$ crosses integer values. Even with oracle choice of $m$, spectrum type determines fundamental accuracy limits.
• Figure 4: Bias-variance trade-off under noisy trace powers. We add multiplicative noise $\hat{p}_k = p_k(1 + \varepsilon_k)$ with i.i.d. $\varepsilon_k \sim \mathcal{N}(0, \eta^2)$ to trace powers for a geometric spectrum ($n = 1024$, $\kappa = 100$) and report RMSE in $K'(0)$ over 1000 trials. Solid curves show $k_{0:m}$ estimators for $m = 3, 4, 5, 6$; dotted horizontal lines mark noise-free interpolation bias $|b_m|$. At low noise ($\eta < 0.5\%$), higher $m$ reduces error by decreasing bias. At high noise ($\eta > 5\%$), higher $m$ increases error due to exponential noise amplification $\alpha_m \sim 2^m/m^{5/4}$. The crossover occurs near $\eta_* = |b_m|/\alpha_m$. For typical Hutchinson noise ($\eta \approx 1$--$5\%$), $m = 4$ provides the optimal bias-variance trade-off.
• Figure 5: Relative error of $k_{0:m}$ estimators across six spectrum types ($n = 1024$). Smooth spectra (geometric, uniform) permit accurate estimation with moderate $m$; lognormal shows larger errors; outlier-dominated spectra (two-point, bimodal, clustered) cause rapid error growth regardless of order.

Theorems & Definitions (52)

• Theorem 3.1: Analytic transform reduction
• proof
• Theorem 4.1: Lagrange derivative weights
• proof
• Remark 4.2
• Proposition 4.3: Weight magnitude growth
• Proposition 5.1: Polynomial exactness
• proof
• Theorem 5.2: Lognormal exactness
• proof