An arXiv preprint reports a theoretical way to estimate the traces of positive trace-class operators in infinite-dimensional settings with a more favorable asymptotic dependence on accuracy than a simpler estimator. At a fixed failure probability, idealized infHutch requires a sample count that grows like one over relative accuracy squared, while idealized infHutch++ grows like one over relative accuracy. In practical terms, the gap becomes more important when the target error is tightened. The document is version v1 of an arXiv preprint dated 28 Aug 2026.
The calculation behind the claim
The study develops infinite-dimensional analogues of Girard-Hutchinson and Hutch++ for positive trace-class operators on separable Hilbert spaces. These are mathematical operators, a generalization of matrices, acting in spaces that can have infinitely many basis directions. Trace-class describes the setting in which the trace being estimated is finite. The paper asks whether random samples can estimate that trace while retaining unbiasedness and high-probability error control.
In idealized infHutch, the algorithm draws independent Gaussian random elements with covariance set by the target operator. It estimates the trace by averaging the squared Hilbert-space norm of each draw. The analysis says this estimate is unbiased for the target trace, meaning its expectation equals the quantity it is meant to recover.
A two-stage estimator
InfHutch++ divides its budget into two independent stages. It uses m/3 samples from a Gaussian distribution with covariance given by A squared for a randomized range finder, which forms an orthonormal basis Q, a set of mutually perpendicular unit directions. It then takes a fresh m/3 samples from a Gaussian distribution with covariance A to estimate the residual trace. The split is the main structural difference from the one-stage average used by infHutch.
Under the theorem's stated conditions, the high-probability guarantee says the infHutch++ estimate lies within a relative tolerance called epsilon of the target trace with probability at least one minus delta. Epsilon is the allowed relative error, while delta is the failure-probability parameter. The authors pair that guarantee with the claim that idealized infHutch++ needs a number of samples that grows like one over the requested accuracy, compared with one over accuracy squared for infHutch at fixed failure probability.
The finite-basis compromise
To make the construction finite, truncated infHutch restricts the Gaussian inputs to a finite-dimensional basis. That introduces truncation bias, and the paper describes the resulting estimator as equivalent to Girard-Hutchinson applied to a compressed version of the operator. In other words, the finite calculation can differ from the full operator not only because of random sampling but also because the chosen basis leaves part of the operator outside the computation.
For truncated infHutch++, Proposition 9 gives a limiting result: with the sample count divisible by 3 and held fixed, the truncated estimator converges in distribution to the idealized infHutch++ estimator, and its expectation converges to the target trace as the truncation is enlarged. The practical, unprojected range finder used in the finite implementation differs from the one analyzed for the idealized method. For that reason, the paper supplies no finite-n Hutch++ error bound for the practical version.
What the examples showed
The numerical evaluation covered two reconstructed examples and a third rational spectral-filter application targeting an eigenvalue count in an interval.
In the sinc-kernel settings, 104 samples were used and relative errors were averaged over 100 runs. For truncated infHutch, the listed errors were 9.6905 × 10⁻² at basis size 10, 3.1967 × 10⁻² at basis size 30, and 6.9704 × 10⁻³ at basis size 100. The listed ContHutch values were 1.8174 × 10⁻¹, 1.0219 × 10⁻¹, and 4.2606 × 10⁻² at settings 0.1, 0.05, and 0.025. The truncated values were lower in the reported rows, although the rows were not matched-accuracy pairs.
A density-of-states test used interval half-length 150, kernel orders 2 and 6, smoothing parameter 0.2, and evaluation point 1. After tuning ContHutch++, the methods achieved similar accuracy in the reported settings. The listed internal chebfun matrix sizes, a measure of the finite discretization used inside the calculation, were 512 and 724 for infHutch++ at basis dimensions 128 and 512, compared with 4096 for the listed ContHutch++ settings.
In the five-well Dirac case, the filtered-trace reference was 52.006128204243. With 80 samples per Hutch++ stage, the reported estimate was 52.0061 plus or minus 0.0002. At a lower budget, a Legendre degree of 768 with 10 samples per stage had average error below 2 relative to the filtered trace near the gap count of 52.
The eleven-well result exposed an important distinction between a filtered estimate and the count being used as a reference. The full gap count was 61, but the chosen contour counted 60 because an eigenvalue of about 0.99593 lay outside the contour ending at plus or minus 0.99. The filtered-trace reference was 60.024852949199. With 80 samples, all four methods reported 60.02485 plus or minus 0.00001. The result tracked the chosen contour count, not the full gap count.
A result with a clear boundary
The paper's strongest claims are split between an idealized theorem and a more conditional computational picture. The sample-complexity orders and high-probability guarantee apply to the idealized algorithms. Finite-basis truncation can introduce bias for infHutch, and the practical unprojected infHutch++ implementation lacks a finite-n Hutch++ error bound.
Paper data and sources
Original title: Stochastic trace estimation for positive trace-class operators
Authors: Zvonimir Bujanović, Luka Grubišić, Daniel Kressner, Hrvoje Olić
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text