GeMA: Learning Latent Manifold Frontiers for Benchmarking Complex Systems

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 (5)

• Figure 1: $\textsc{ProMan-VAE}$ 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.
• 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.
• 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 $\textsc{GeMA}$ 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 $\textsc{GeMA}$, 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.
• 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 $\textsc{GeMA}$ 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.

Theorems & Definitions (1)

• Definition 3.1: Certification radius