GES Algorithm: Causal, Robotic & 3D Splatting

Updated 18 May 2026

GES is a collection of algorithms including Greedy Equivalence Search for causal inference, geometry‐based expert selection in robotics, and generalized exponential splatting for 3D rendering.
The causal GES variant utilizes forward and backward search phases with score-based or neural conditional dependence measures to optimally recover Markov equivalence classes.
Its robotics and rendering adaptations employ clustering with expert gating and GEF splats with modulated loss, achieving impressive performance and efficiency in their respective domains.

The acronym GES denotes several context-dependent algorithms in recent literature, each foundational to its respective field. The principal usages include: Greedy Equivalence Search for causal graphical model discovery, Geometry-based Expert Selection in robotic dexterous grasping, and Generalized Exponential Splatting in differentiable 3D scene rendering. This article provides comprehensive exposition of the three main algorithms established under the name GES, with precise attention to their mathematical structure and performance characteristics.

1. Greedy Equivalence Search (GES) for Causal Structure Learning

The canonical GES algorithm, originally articulated by Chickering (2002) and extensively developed in subsequent literature, is a score-based method for discovery of a Markov equivalence class (MEC) of directed acyclic graphs (DAGs) representing the causal structure underlying i.i.d. data on variables $V = \{X_1, \dots, X_d\}$ . The output is a Completed Partially Directed Acyclic Graph (CPDAG) encoding all DAGs Markov equivalent to the true causal DAG.

GES proceeds in two phases:

Forward Equivalence Search (FES): Starting from the empty CPDAG, iteratively add the edge whose addition yields maximal improvement in a global score (commonly BIC, BDeu), halting when no further addition increases score.
Backward Equivalence Search (BES): From the CPDAG produced by FES, iteratively remove the edge whose deletion most improves the score, continuing until no deletion is beneficial.

Let $S(G)$ denote the decomposable score for DAG $G$ , typically based on local conditional likelihood penalized for complexity. Consistency in recovering the true MEC is guaranteed under the assumptions of causal Markov and faithfulness if the score is locally consistent; i.e., in the large sample regime, adding (deleting) an edge improves the score if and only if the corresponding conditional dependence (independence) actually holds in $P_V$ . GES is robust to various data properties, but suffers performance drops under substantial model misspecification or strong nonlinearities (Shen et al., 2022, Ramsey, 2015).

GES can be augmented with the nonparanormal transform for handling variables that are monotonic transforms of Gaussians, enhancing its effectiveness under moderate non-Gaussianity and nonlinearity (Ramsey, 2015). For very strong nonlinear dependencies, hybrid procedures (PC→GES) may perform better.

2. Conditional Dependence-Guided Reframing and the Neural Conditional Dependence (NCD) GES

The original GES formalism can be reframed by recognizing that only local conditional (in)dependence decisions are necessary at each step. Any $\tau$ -consistent family of conditional dependence measures $T_n(X,Y|Z)$ can substitute score differences: $T_n < \tau$ indicates independence $X \perp Y|Z$ , while $T_n > \tau$ signals dependence.

This leads to the following reformulation:

Forward phase: Insert the edge $(X_i, X_j)$ with maximal $S(G)$ 0 as long as $S(G)$ 1.
Backward phase: Delete the edge with minimal $S(G)$ 2, as long as $S(G)$ 3.

A family $S(G)$ 4 is $S(G)$ 5-consistent if:

For all $S(G)$ 6, $S(G)$ 7 as $S(G)$ 8 iff $S(G)$ 9.
Otherwise, $G$ 0.

The reframed GES retains optimality under the standard Markov and faithfulness assumptions, provided $G$ 1-consistency holds (Shen et al., 2022).

Neural Conditional Dependence (NCD) Measure

The Neural Conditional Dependence measure leverages deep neural networks to construct expressive function classes for conditional independence testing. The measure $G$ 2 is the maximal squared Pearson correlation of centered functions $G$ 3, $G$ 4 over all $G$ 5 annihilated in mean given $G$ 6:

$G$ 7

Empirically, $G$ 8 is estimated by alternating between descent on regression losses to center $G$ 9, $P_V$ 0, and ascent on the squared sample correlation. All function estimators are multilayer perceptrons with spectral normalization to enforce Lipschitz constraints.

Under certain regularity conditions, $P_V$ 1 in probability and the procedure is a consistent learner for the true MEC (Shen et al., 2022).

Experimental results demonstrate:

Superior performance of reframed GES + NCD in terms of Structural Hamming Distance (SHD), Structural Intervention Distance (SID), and F1 score, especially versus kernel and parametric alternatives.
Robustness to model misspecification and scalability to larger datasets and graphs.
Significant computational advantage compared to kernel-based measures (e.g., KGV), with NCD scaling linearly in $P_V$ 2 versus hours for kernel approaches at $P_V$ 3 examples.

3. Geometry-Based Expert Selection (GES) in Dexterous Robotic Grasping

A distinct usage, Geometry-based Expert Selection, arises in dexterous robotic manipulation, specifically in the GES-UniGrasp framework (Xu et al., 28 Sep 2025). Here, GES denotes a two-stage scheme for selecting specialized reinforcement learning (RL) expert policies based on object geometry:

Offline Expert Preparation: Objects are embedded via a CurveNet-style network from point clouds to $P_V$ 4, then clustered (K-means) into groups by geometric similarity. Each cluster receives a dedicated RL grasping expert trained via PPO, with an additional expert for “hard” objects inadequately handled by the main clusters.
Online Expert Selection: For a novel object, the same embedding is computed and a gating MLP network sends softmax scores over experts, choosing the most probable expert to execute the grasp (pre-grasp pose computed via contact synthesis and joint-space retargeting).

This modular expert selection confers robust generalization. On a 773-object, 82-category benchmark, GES-UniGrasp attains train/test success rates of 99.4%/96.3%, outperforming single-expert and naive mixture-of-expert baselines (Xu et al., 28 Sep 2025).

4. Generalized Exponential Splatting (GES) for 3D Radiance Fields

A third, entirely unrelated GES algorithm emerges in the context of 3D scene rendering. Generalized Exponential Splatting employs the Generalized Exponential Function (GEF):

$P_V$ 5

GEF generalizes the Gaussian ( $P_V$ 6) and Laplace ( $P_V$ 7). In GES, each 3D “splat” is a GEF characterized by position, covariance, sharpness $P_V$ 8, opacity, and color coefficients. Rendering applies EWA splatting and volume integration, with the rendered image computed as an integral over rays through GEFs, with explicit shape parameter modulation.

Optimization uses a frequency-modulated multi-term loss, with a novel component $P_V$ 9 focusing learning on coarse-to-fine image structure. GES demonstrates improved representation of signals with sharp edges compared to Gaussian splatting, significant memory reduction (49%–46% less than Gaussian baselines), and up to 39% faster rendering, with matched or improved SSIM and PSNR on Mip-NeRF360, Tanks & Temples, and Deep Blending datasets (Hamdi et al., 2024).

5. Practical Considerations and Limitations

GES for Causal Discovery

Parametric vs. Nonparametric: Traditional GES is optimal for (faithful) linear-Gaussian settings. Nonparametric reframings provide robustness to nonlinearity but introduce hyperparameter sensitivity.
Conditional Dependence Estimators: Choice of estimator is application-specific; NCD is powerful but computationally heavier than closed-form rank-based or kernel alternatives.
Sample Complexity: Neural estimators for dependence demand sufficient sample size for reliable ranking of local moves. Network architecture must be tuned (typically 2–4 layers, 20–100 neurons are robust).
Transform Preprocessing: The nonparanormal transform is recommended if variables appear to be monotone transforms of Gaussians; it does not harm GES performance even if unnecessary (Ramsey, 2015).
Hybridization: For highly nonlinear/strongly non-Gaussian scenarios, hybrid two-stage approaches (e.g., PC followed by GES) may provide more accurate skeletons.

Geometry-Based Expert Selection

Clustering Quality: The gating network's performance depends on the separability of geometric features and the quality of initial CurveNet embedding.
Policy Generalization: Success rates remain high on novel instances and unseen categories, but remain contingent on sufficient geometric coverage during training.

Exponential Splatting

Component Count: GES achieves comparable or better rendering with half the number of splats used by Gaussian baselines.
Sharpness Control: The adaptation of $\tau$ 0 enables sharp transitions, but necessitates careful management during optimization to avoid artifacts.
Optimization Stability: Frequency-modulated losses accelerate convergence but may require empirical adjustment of hyperparameters for stability across datasets.

6. Directions for Extension and Open Problems

Causal Discovery: Embedding NCD or other $\tau$ 1-consistent estimators into differentiable relaxations of the search (e.g., NOTEARS-style) offers possible speedups at the cost of global optimality.
Alternative Dependence Measures: Any estimator with $\tau$ 2-consistency can substitute for NCD in reframed GES; systematic benchmarking is an open avenue (Shen et al., 2022).
Do-Calculus Integration: For interventional data, combining GES with explicit do-calculus reasoning could yield refined equivalence class identification.
GES in Robotics: Scaling the number of experts, adopting more expressive shape features, or enabling continual learning remain open research trajectories.
Splatting: Improvements in adaptive split/prune mechanisms and exploration of nonparametric splat compositions could further enhance signal fidelity and resource efficiency.

7. Summary Table of GES Algorithmic Variants

Domain	GES Expansion	Core Mechanism	Key Reference
Causal Inference	Greedy Equivalence Search	Score/local-dependence search	(Shen et al., 2022, Ramsey, 2015)
Robotics	Geometry-based Expert Selection	Clustering + expert gating	(Xu et al., 28 Sep 2025)
3D Rendering	Generalized Exponential Splatting	GEF splats + modulated loss	(Hamdi et al., 2024)

Each “GES” algorithm achieves state-of-the-art performance within its original context, with methods unified more by the greedy or expert-selection paradigm than mathematical lineage. For detailed formulations, network architectures, and empirical benchmarks, refer to the primary arXiv references provided.