Sustainable Harmonic Mean (FMS) Overview

• Sustainable Harmonic Mean (FMS) is a framework that uses harmonic-mean constructions to enforce a balanced aggregation of data by emphasizing lower values to avoid distortion by outliers.
• It is applied across diverse fields such as scientometrics, reinforcement learning, matrix analysis, Bayesian computation, and sensor fusion to achieve robust averaging.
• Key methodologies include divergence minimization, modified harmonic operators, and learned density estimation techniques that maintain sustainability under various operational challenges.

Searching arXiv for the cited work and closely related harmonic-mean/FMS literature. Sustainable Harmonic Mean, abbreviated here as FMS when referring to the “sustainable harmonic mean/FMS mean” terminology appearing in the matrix-means literature, designates a family of harmonic-mean-based constructions used to enforce balanced aggregation under asymmetry, dependence, or uncertainty. Across the cited literature, the harmonic mean is repeatedly used when a quantity should not be inflated by one large component while another remains small: in scientometrics it combines the number of papers and average citations per paper into a single-value index; in average-reward reinforcement learning it is presented as the correct averaging operator for rates with varying durations; in matrix analysis it appears as a barycentric limit associated with log-determinant divergence and is explicitly linked to the “sustainable harmonic mean/FMS mean”; in Bayesian computation it underlies evidence estimators whose practical viability depends on variance control; and in distributed tracking it yields a harmonic mean density interpolation positioned between naive fusion and over-conservative alternatives (Germano, 2020, Shtossel et al., 6 May 2026, Kim et al., 2024, McEwen et al., 2021, Sharma et al., 2024).

1. Terminological scope and core idea

The most explicit occurrence of FMS in the provided corpus is in the matrix-analysis setting, where the paper on the multivariable mean $G_t(\omega; \mathbb{A})$ states that “the FMS mean (as defined by Chebbi and Moahker [8]) is the barycenter/minimizer of a log-determinant divergence,” and further identifies $G_t$ as characterizing this structure for general $n$, weights, and parameter $t$ (Kim et al., 2024). In that usage, FMS is not merely a scalar average but a variationally defined barycenter on positive definite matrices.

A broader cross-domain reading is also supported by the data. In each domain, the harmonic mean is selected because it weighs low values heavily, and therefore rewards balance rather than allowing one dominant term to conceal weakness in another. In the scientometric proposal, this appears as a balance between productivity and average impact (Germano, 2020). In SMDP reinforcement learning, it appears as direct averaging of reward rates rather than a ratio of separate averages, especially when rewards and durations are dependent or non-stationary (Shtossel et al., 6 May 2026). In sensor fusion, harmonic mean density interpolation is presented as a conservative but less over-dispersed alternative to arithmetic pooling (Sharma et al., 2024). This suggests that “sustainable” in FMS is tied to robustness against pathological extremes rather than to a single universal formula.

2. Scalar harmonic-mean indices in research evaluation

A direct scalar instantiation is given in “The h-index and the Harmonic Mean” (Germano, 2020). The paper proposes the harmonic mean between the number of papers $N_p$ and the average number of citations per paper $N_c$ as a single-value index for quantifying research output: $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$ Here $N_c$ is the average number of citations per paper, with

$N_c = N_{c,\mathrm{tot}}/N_p.$

The paper positions this HM index against the Hirsch $h$-index, defined by

$G_t$0

According to the summary, both $G_t$1 and $G_t$2 penalize “outrageous” cases in which the same total citation count is distributed very unevenly across $G_t$3 and $G_t$4, but the HM-index tends to be “less disposed to reward the middle” (Germano, 2020). In a realistic comparison involving Germano, Piomelli, Moin, and Cabot, the HM-index gives higher scores to some high-impact/low-output profiles that are penalized by the $G_t$5-index, because it reflects combined productivity and average impact rather than the citation threshold structure of $G_t$6 alone (Germano, 2020).

The paper attributes three principal advantages to this construction: simplicity, balance across extremes, and greater generosity than $G_t$7 in some mixed productivity-impact cases. It also records several limitations: sensitivity when either $G_t$8 or $G_t$9 is very low, possible overvaluation of more balanced moderate profiles, and insensitivity to distributional skewness such as a few highly cited papers alongside many uncited ones (Germano, 2020). The sustainability interpretation is stated directly: the harmonic mean “naturally encourages researchers to maintain not just quantity or quality, but a sustainable balance between both.” The same source adds that, if extended or refined “as a 'Sustainable Harmonic Mean Index' or FMS,” it could become the basis of a more robust and fair quantitative evaluation framework (Germano, 2020).

3. Rate averaging, robustness, and modified harmonic means in SMDPs

A second major formulation appears in average-reward reinforcement learning for semi-Markov decision processes. “A Harmonic Mean Formulation of Average Reward Reinforcement Learning in SMDPs” argues that the harmonic mean is the mathematically correct operator for averaging stepwise rates when action durations vary (Shtossel et al., 6 May 2026). Existing algorithms such as SMART and Relaxed-SMART are summarized as approximating the average reward rate either by the ratio of cumulative rewards to cumulative times or by the ratio of exponential moving averages of rewards and durations. The paper states that these methods are theoretically sound only if rewards and durations are stationary and/or independent; under dependence or non-stationarity they can be inaccurate (Shtossel et al., 6 May 2026).

For stepwise rates $n$0, the classical harmonic mean is written as

$n$1

when all $n$2. The theoretical distinction from the ratio-of-means estimator is formalized in the condition

$n$3

where

$n$4

The paper therefore treats the harmonic mean not as a heuristic but as the correct averaging operator for rates under variable durations (Shtossel et al., 6 May 2026).

Because standard harmonic means are undefined for zero rewards and pathological for mixed-sign rewards, the paper introduces a modified harmonic mean operator $n$5. For a sequence $n$6 partitioned into positive, negative, and zero subsets $n$7, it defines

$n$8

equivalently

$n$9

The paper states that this operator is general, internal, idempotent, symmetric, and monotone, though not quasi-arithmetic (Shtossel et al., 6 May 2026). It is then used in the algorithm $t$0, which tracks separate EMA statistics for positive, negative, and zero-reward reciprocals and uses the resulting harmonic estimate as the running average reward rate.

The sustainability language in this paper is explicit: by avoiding over-estimation under non-stationarity, heavy tails, or regime shifts, the harmonic approach is said to support “more sustainable, robust, and accurate” reinforcement learning in open-ended environments (Shtossel et al., 6 May 2026). A plausible implication is that FMS-type harmonic constructions gain their “sustainable” character when they remain valid under coupling and distributional shift rather than only under iid-like assumptions.

4. Matrix-valued FMS means and barycentric structure

The most technically specific FMS-related construction in the data is the new multivariable matrix mean $t$1 introduced in “New multivariable mean from nonlinear matrix equation associated to the harmonic mean” (Kim et al., 2024). For positive definite matrices $t$2, weights $t$3, and $t$4, the mean is defined as the unique positive definite solution of

$t$5

This family interpolates continuously between the weighted arithmetic and weighted harmonic means. At $t$6,

$t$7

while as $t$8,

$t$9

Moreover,

$N_p$0

and for $N_p$1,

$N_p$2

The paper proves uniqueness by showing the defining map is a contraction in the Thompson metric, and it lists idempotency, homogeneity, permutation invariance, monotonicity, continuity, congruence invariance, self-duality, and arithmetic-harmonic inequalities among the core properties (Kim et al., 2024).

The FMS connection enters through divergence geometry. The paper states that $N_p$3 is the right mean or barycenter for the log-determinant $N_p$4-divergence

$N_p$5

with $N_p$6. It further states that at $N_p$7, equivalently $N_p$8, the divergence reduces to the “harmonic divergence,” so $N_p$9 is the barycenter for this harmonic-like divergence and thereby relates to the sustainable harmonic mean/FMS mean (Kim et al., 2024).

The following summary captures the interpolation structure already stated in the source:

Parameter regime	Mean
$N_c$0	Arithmetic mean
$N_c$1	Interpolating mean
$N_c$2	Harmonic mean

Within this literature, FMS is therefore best understood as a barycentric or variational harmonic mean on positive definite matrices, embedded in a divergence-minimization framework rather than merely as a scalar average (Kim et al., 2024).

5. Sustainable harmonic formulations in Bayesian evidence estimation

The harmonic mean also appears in Bayesian computation, though here the central issue is not aggregation balance alone but estimator stability. “Machine learning assisted Bayesian model comparison: learnt harmonic mean estimator” revisits the classic harmonic mean estimator for the marginal likelihood $N_c$3, defined through posterior samples $N_c$4 by

$N_c$5

The paper states that the classical estimator can have extremely large or even infinite variance because it can be interpreted as importance sampling with the prior as target and posterior as proposal, so tail mismatch causes unstable weights (McEwen et al., 2021).

Its remedy is the learnt harmonic mean estimator (LHME), which introduces a learned normalized target density $N_c$6: $N_c$7 The inaccessible zero-variance optimum is

$N_c$8

The paper states that learning $N_c$9 from posterior samples resolves the large-variance problem while remaining agnostic to the posterior sampling strategy (McEwen et al., 2021). The normalizing-flow extension in “Learned harmonic mean estimation of the Bayesian evidence with normalizing flows” replaces earlier machine-learning models by a flow-based density, then concentrates it via a temperature parameter $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$0 to enforce thinner tails: $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$1 That paper states that the learned estimator with normalizing flows is robust, scalable, flexible, and agrees with ground-truth or nested-sampling estimates on the reported benchmarks (Polanska et al., 2024).

The connection to sustainability is explicit in both summaries. The older work frames the original harmonic mean estimator as “typically unsustainable” because of variance blow-up, and the learned version as making harmonic-mean-based evidence estimation sustainable by variance control (McEwen et al., 2021). The normalizing-flow follow-up describes the method as reliable because it prevents catastrophic tail failures while remaining usable with saved MCMC chains or variational samples (Polanska et al., 2024). This suggests that, in Bayesian computation, a sustainable harmonic mean is one whose asymptotic identity is preserved but whose operational form is regularized enough to remain finite-variance and diagnostically tractable.

6. Harmonic mean density in conservative track fusion

A further extension appears in distributed sensor fusion through the harmonic mean density (HMD) interpolation introduced in “On Pooling-Based Track Fusion Strategies : Harmonic Mean Density” (Sharma et al., 2024). Given local densities $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$2 and $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$3, the unnormalized HMD interpolation is

$HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$4

and the normalized density is

$HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$5

with

$HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$6

The paper situates HMD between covariance intersection, which is geometric-mean-density-based and conservative but cumbersome for Gaussian mixtures, and arithmetic mean density, which is easy to compute but often over-conservative (Sharma et al., 2024). HMD is presented as handling both Gaussian and Gaussian-mixture densities within nearly the same framework, while avoiding the non-integer power operations required by generalized covariance intersection on mixtures.

Several mathematical properties are listed. The normalization constant is convex in $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$7 and less than $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$8 for $HM = H(N_p, N_c) = \frac{2N_p N_c}{N_p + N_c}.$9. The fused density obeys the pointwise bounds

$N_c$0

and satisfies

$N_c$1

The simulations reported in 2D and 3D maneuvering-target scenarios suggest lower RMSE than other conservative fusion strategies while remaining consistent (Sharma et al., 2024).

The sustainability interpretation here is operational and network-level: HMD is described as robust to unknown cross-correlation, avoiding both overconfidence from naive fusion and over-conservativeness from arithmetic pooling, and therefore supporting “sustainable” fusion in decentralized, communication-limited, or unreliable sensor networks (Sharma et al., 2024). In this setting, a sustainable harmonic mean is not a scalar score but a pooling rule whose uncertainty remains controlled over repeated fusion cycles.

7. Conceptual unification and limitations

Across the cited works, three recurring structural themes define the sustainable harmonic mean idea. First, the harmonic mean is used when low-valued components must exert disproportionate influence, preventing inflation by one large component alone. This is explicit in research evaluation, where a high $N_c$2 cannot compensate completely for a low $N_c$3, or vice versa (Germano, 2020). Second, harmonic formulations are preferred when the object being averaged is intrinsically rate-like or reciprocal in structure, as in reward-per-time optimization and evidence reciprocals (Shtossel et al., 6 May 2026, McEwen et al., 2021). Third, in matrix and density settings, harmonic constructions often appear as barycenters or pooling operators constrained by divergence geometry or conservative fusion requirements (Kim et al., 2024, Sharma et al., 2024).

At the same time, the literature also delineates the limits of the approach. In scientometrics, a single harmonic index ignores citation-distribution skewness (Germano, 2020). In reinforcement learning, the classical harmonic mean must be modified to handle zero and mixed-sign rewards (Shtossel et al., 6 May 2026). In Bayesian evidence computation, the naive harmonic mean estimator is unusable without variance control (McEwen et al., 2021). In track fusion, HMD remains an approximation framework whose denominator may require Gaussian approximation for tractability on mixtures (Sharma et al., 2024). The common lesson is that sustainability does not arise from the harmonic mean alone; it depends on whether the surrounding formulation preserves internality, robustness, boundedness, or consistency under the domain’s specific pathologies.

Taken together, these works support a precise encyclopedic characterization: Sustainable Harmonic Mean (FMS) denotes a class of harmonic-mean-centered constructions that are designed to remain meaningful under imbalance, dependence, or uncertainty, and that often acquire their full technical significance only when embedded in an extended framework such as divergence minimization, modified mean operators, learned importance targets, or conservative density pooling (Kim et al., 2024, Shtossel et al., 6 May 2026, Polanska et al., 2024).