GeMA: Learning Latent Manifold Frontiers for Benchmarking Complex Systems

Jia Ming Li; Anupriya; Daniel J. Graham

cs.LGMar 17, 2026

GeMA: Learning Latent Manifold Frontiers for Benchmarking Complex Systems

Authors: Jia Ming Li, Anupriya, Daniel J. Graham

Organizations: Transport Strategy Centre, Imperial College London, London, SW7 2AZ, United Kingdom.

Abstract

Benchmarking the performance of complex systems such as rail networks, renewable generation assets and national economies is central to transport planning, regulation and macroeconomic analysis. Classical frontier methods, notably Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA), estimate an efficient frontier in the observed input-output space and define efficiency as distance to this frontier, but rely on restrictive assumptions on the production set and only indirectly address heterogeneity and scale effects. We propose Geometric Manifold Analysis (GeMA), a latent manifold frontier framework implemented via a productivity-manifold variational autoencoder (ProMan-VAE). Instead of specifying a frontier function in the observed space, GeMA represents the production set as the boundary of a low-dimensional manifold embedded in the joint input-output space. A split-head encoder learns latent variables that capture technological structure and operational inefficiency. Efficiency is evaluated with respect to the learned manifold, endogenous peer groups arise as clusters in latent technology space, a quotient construction supports scale-invariant benchmarking, and a local certification radius, derived from the decoder Jacobian and a Lipschitz bound, quantifies the geometric robustness of efficiency scores. We validate GeMA on synthetic data with non-convex frontiers, heterogeneous technologies and scale bias, and on four real-world case studies: global urban rail systems (COMET), British rail operators (ORR), national economies (Penn World Table) and a high-frequency wind-farm dataset. Across these domains GeMA behaves comparably to established methods when classical assumptions hold, and provides additional insight in settings with pronounced heterogeneity, non-convexity or size-related bias.

Figures & tables

Figure 1: ProMan-VAE architecture. A split-head encoder maps observed inputs and outputs $(\mathbf{x},\mathbf{y})$ to latent technology $(\mathbf{z})$ and inefficiency $(u)$ . The decoder $\mathcal{G}_{\theta}$ reconstructs the frontier output from $(\mathbf{x},\mathbf{z})$ , and the realised output is obtained by scaling the frontier with $\exp(-u)$ . A monotonicity regulariser encourages the decoder to be weakly increasing in each input dimension.

Percentile	p0	p25	p50	p75	p95
$R_{\mathrm{cert}}$	0.105	0.227	0.293	0.340	0.392

Table 1: ORR: certification radius percentiles across operator–year observations.

Panel of Figure 2 — Figure 2: (Left) Distribution of certification radii $R_{\mathrm{cert}}(\mathbf{x}_{i})$ across operator–year observations. A visible left tail indicates cases where performance assessments rely on locally ill-conditioned frontier geometry. (Right) Model-based score versus certification radius. The dashed lines indicate the top decile of the score and the bottom quartile of $R_{\mathrm{cert}}$ , highlighting observations with high point performance but weak robustness guarantees.

Panel of Figure 3 — Figure 3: Wind farms: learned frontiers vs specification-based toy curves for two representative farms (Farm 2 and 5). Scatter points show observed capacity factor versus hub-height wind speed; red curves show the GeMA frontier (normalised by its 95th percentile per farm); blue curves show simple toy turbine curves constructed from manufacturer cut-in, rated and cut-out speeds. Orange markers indicate specification-based operating points. (Left) Frontier learned without using the turbine threshold (cut-in/rated speeds) as inputs. (Right) Frontier learned with the turbine threshold (cut-in/rated speeds) included as additional inputs, yielding a tighter alignment with the theoretical plateau and a more pronounced decline near cut-out.

Figure 4: COMET: UMAP embeddings of latent technology vectors $\mathbf{z}_{i}$ learned by GeMA , coloured by GMM cluster assignment ( $k=4$ ). (a) Two-dimensional UMAP projection of the latent technology space used for visualising peer groups. (b) Three-dimensional UMAP view of the same latent manifold, illustrating its overall geometry; the projected points in (a) correspond to this surface. (c) DMUs whose outputs lie on the estimated frontier, shown as the subset of points mapped to the boundary of the learned manifold in latent space. These boundary points represent units operating at the highest efficiency given their latent technology, i.e. on the manifold frontier.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Scenario A	Scenario B	Scenario C
	Frontier error / Rank corr.	ARI / Rank corr.	Size corr. / Rank corr.
DEA (VRS)	- / 0.655 (0.042)	0.782 (0.024)	- / 0.801 (0.023)
SFA (Translog)	0.216 (0.009) / 0.727 (0.041)	0.803 (0.017)	0.038 (0.022) / 0.818 (0.014)
FDH	- / 0.595 (0.039)	0.701 (0.028)	- / 0.726 (0.027)
CNLS	10.395 (38.537) / 0.350 (0.041)	0.506 (0.039)	0.154 (0.064) / 0.786 (0.056)
ML predictor (RF)	0.405 (0.719) / 0.780 (0.023)	0.767 (0.028)	0.093 (0.024) / 0.778 (0.023)

Appendix

Table 2: Synthetic experiments: Monte Carlo summary of frontier approximation error and rank correlation with true inefficiency across methods and scenarios. Means with standard deviations in parentheses.

Name	Description	Role	Transformation
Operator	Metro system identifier	Entity ID	–
Year	Calendar year	Time ID	–
Staff	Number of staff (FTE)	Input	$\log(1+x)$
Capacity	Rolling stock seating capacity (car seats)	Input	$\log(1+x)$
Stations	Number of stations	Input	$\log(1+x)$
RouteLength	Network Length (km)	Input/Scale	$\log(1+x)$

Appendix

Table 3: COMET variables: definitions and roles.

Name	Description	Role	Transformation
Operator	Train operating company ID	Entity ID	–
Year	Calendar year	Time ID	–
Labour	Number of staff (FTE)	Input	$\log(1+x)$
Route	Route kilometres operated by operators	Input/Scale	$\log(1+x)$
Stock	Train rollingstock in service	Input	$\log(1+x)$
Plan	Numbers of trains planned	Input	$\log(1+x)$

Appendix

Table 4: ORR variables: definitions and roles.

Name	Description	Role	Transformation
country	Country code / identifier	Entity ID	–
year	Calendar year	Time ID	–
rkna	Capital services at constant 2017 national prices	Input	$\log(\text{rkna}/\text{emp})$
hc	Human capital index	Input	$\log(\text{hc})$
emp	Number of persons engaged (millions)	Input	$\log(\text{emp})$
rgdpo	Output-side real GDP at chained PPPs (million)	Output	$\log(\text{rgdpo}/\text{hours})$

Appendix

Table 5: PWT variables: definitions and roles.

Name	Description	Role	Transformation
asset_num	Wind farm identifier	Entity ID	–
year	Calendar year	Time ID	–
ws_hub	Hub-height wind speed	Input	$\log(1+x)$
temp_air	Air temperature	Input	$\log(1+x)$
pressure_air	Air pressure	Input	$\log(1+x)$
rel_humidity	Relative humidity	Input	$\log(1+x)$

Appendix

Table 6: WF variables: definitions and roles.

Domain	$K$ (tech dim)	Hidden dim	Epochs	Batch size	Learning rate	$\gamma_{u}$
WF	2	192	300	1024	$5\times 10^{-4}$	1.0
PWT	4	192	200	256	$3.8\times 10^{-3}$	0.213
ORR	4	128	200	1792	$4.7\times 10^{-3}$	0.223
COMET	2	128	200	2048	$4.4\times 10^{-3}$	0.201

Appendix

Table 7: ProMan-VAE hyperparameters by domain.

Domain	Activation	Spectral norm	Mono weight	Ranking loss weight
WF	SILU	No	0.0	(if used; else 0)
PWT	SILU	(default)	$4.8\times 10^{-4}$	0.271
ORR	GELU	Yes	$2.2\times 10^{-4}$	0.298
COMET	GELU	Yes	$1.0\times 10^{-4}$	0.056

Appendix

Table 8: Additional architectural and regularisation choices.

Farm	Specification (capacity-weighted)			GeMA frontier (no thresholds)		GeMA frontier (with thresholds)
Site	$v_{c}^{\mathrm{spec}}$	$v_{r}^{\mathrm{spec}}$	$v_{co}^{\mathrm{spec}}$	$v_{c}^{\mathrm{fit}}$	$v_{r}^{\mathrm{fit}}$	$v_{c}^{\mathrm{fit}}$	$v_{r}^{\mathrm{fit}}$
	(m/s)	(m/s)	(m/s)	(m/s)	(m/s)	(m/s)	(m/s)
1	3.0	10.1	22.7	0.9	15.1	0.7	15.1
2	2.5	10.5	20.0	0.5	12.2	0.5	11.6
3	3.0	10.4	25.0	0.5	15.7	0.5	16.3
4	3.0	9.5	20.0	0.5	15.8	1.3	12.2

Appendix

Table 9: Wind farms: specification-based (capacity-weighted) vs frontier-fitted cut-in and rated wind speeds. Specification values

v_{c}^{\mathrm{spec}},v_{r}^{\mathrm{spec}},v_{co}^{\mathrm{spec}}

are computed from manufacturer data; fitted values

(v_{c}^{\mathrm{fit}},v_{r}^{\mathrm{fit}})

are obtained by least-squares on the learned GeMA frontiers, with and without turbine thresholds as inputs. Fitted parameters are obtained post hoc from the learned frontiers and are not directly constrained during training.

Name	N Cap.	Model	Cap.	Ht	Rotor	#	Cut-in	Cut-out	RatedSpeed	Swt.Ar	PD1	PD2
	(MW)		(kW)	(m)	(m)		(m/s)	(m/s)	(m/s)	(m 2 )	(W/m 2 )	(m 2 /kW)
F1	75	GW1500/87	1500	85.0	87.0	50	3.0	22	9.9	5890	254.7	3.9
F1	24	H93 L-2.0MW	2000	85.5	93.0	12	3.0	25	10.8	6792.9	294.4	3.4
F2	200	GW3000/140	3000	120.0	140.0	67	2.5	20	10.5	15747	193.9	5.2
F3	49.5	UP86-1500	1500	80.0	86.0	33	3.0	25	10.0	5809	258.2	3.9
F3	49.5	UP82-1500	1500	80.0	82.0	33	3.0	25	10.8	5384	278.6	3.6

Appendix

Table 10: Technical specifications of turbine configurations at each wind farm. Cut-in, rated and cut-out speeds are manufacturer-reported thresholds for individual turbine models.

Panel of Figure 5 — Figure 5: Additional wind farm power-curve plots. For each farm, scatter points show observed capacity factor versus hub-height wind speed; red curves show the learned GeMA frontier (normalised by its 95th percentile); blue curves show specification-based “toy” turbine curves constructed from capacity-weighted cut-in, rated and cut-out speeds; orange markers highlight the specification-based operating points. The overall shape and operating points are consistent across farms, with high-wind declines indicating early curtailment where frequent high-wind events occur.

Cluster	# Test obs.	Route length	Stations	Capacity	Staff	Passenger-km	Car-km
A	27	120.7 (231.1)	100.5 (120.5)	40.0 (16.7)	3818 (7980)	3107 (5184)	94.4 (172.0)
B	28	186.4 (266.1)	87.0 (190.3)	43.2 (15.3)	5688 (11720)	5532 (13115)	140.1 (281.8)
C	16	141.2 (135.0)	104.0 (88.3)	40.5 (10.4)	5600 (6703)	6660 (14090)	128.9 (211.5)
D	13	115.1 (116.9)	91.0 (59.0)	44.4 (10.8)	4529 (11844)	6201 (7541)	114.7 (164.3)

Appendix

Table 11: COMET (test period 2018–2019): cluster-wise summaries of observable characteristics. Reported as median (interquartile range) over operator–year observations.

Cluster	# Test obs.	Route length (IQR)	DEA $\hat{\theta}$ (IQR)	GeMA score (IQR)	RF proxy $u$ (IQR)
A	27	120.7 (231.1)	1.000 (0.050)	-0.255 (0.319)	0.110 (0.283)
B	28	186.4 (266.1)	0.996 (0.073)	-0.447 (0.307)	0.075 (0.269)
C	16	141.2 (135.0)	0.995 (0.101)	-0.310 (0.505)	0.043 (0.185)
D	13	115.1 (116.9)	1.000 (0.043)	-0.555 (0.371)	0.005 (0.179)

Appendix

Table 12: COMET (test period 2018–2019): cluster-wise summaries of observable scale and baseline scores. Reported as median (IQR) over operator–year observations.

Cluster	# Test obs.	$\max_{c}p_{ic}$ (IQR)	Share $\max_{c}p_{ic}\geq 0.8$	Share $\max_{c}p_{ic}\geq 0.9$
A	27	0.534 (0.195)	11.1%	0.0%
B	28	0.611 (0.305)	21.4%	7.1%
C	16	0.759 (0.346)	43.8%	18.8%
D	13	0.797 (0.272)	46.2%	23.1%

Appendix

Table 13: COMET (test period 2018–2019): posterior cluster assignment confidence from the GMM on latent technology space. Reported as median (IQR) of

\max_{c}p_{ic}

and the share of observations exceeding common confidence thresholds.

Percentile	Certification radius
0%	0.135
5%	0.177
25%	0.249
50%	0.322
75%	0.364
95%	0.440

Appendix

Table 14: COMET: certification radius percentiles (whitened-input robustness diagnostic).

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