Geometry-Based Expert Selection

Updated 14 July 2026

Geometry-based Expert Selection (GES) is a methodological family where expert choice is driven by explicit geometric, similarity, or dependency signals rather than uniform aggregation.
GES leverages metrics like cosine similarity, sparse precision matrices, and point-cloud clustering to dynamically select and suppress redundant or weak experts.
Its applications span sparse MoE routing, Gaussian-process aggregation, and dexterous grasping, leading to enhanced efficiency and balanced specialization.

Geometry-based Expert Selection (GES) denotes a class of expert-selection schemes in which the choice of expert is driven by an explicit geometry, similarity, or structured dependency signal over inputs, experts, or expert outputs, rather than by uniform aggregation or purely reward-driven switching. In the cited literature, the label appears most directly in dexterous grasping, where object geometry is used to cluster shapes and gate among specialized grasping experts (Xu et al., 28 Sep 2025). Closely related formulations appear in local Gaussian-process approximations, where a sparse precision matrix over expert predictions identifies the most important experts (Jalali et al., 2021), and in sparse Mixture-of-Experts (MoE) routing, where cosine similarity in gating space or hidden-state geometry is used to analyze or modify expert activation (Zheng et al., 15 Oct 2025, Wang et al., 10 Apr 2026). This suggests that GES is better understood as a methodological family than as a single canonical algorithm.

1. Scope and defining characteristics

Across the literature, GES is unified less by a single architecture than by a shared principle: expert choice is conditioned on a geometry-like structure that encodes which experts are relevant, redundant, central, or compatible for the current input. In some settings the geometry is literal object shape; in others it is the geometry of hidden representations, gating vectors, or a dependency graph among expert predictions (Xu et al., 28 Sep 2025, Jalali et al., 2021, Zheng et al., 15 Oct 2025, Wang et al., 10 Apr 2026).

Setting	Geometry signal	Selection action
Sparse MoE routing	Cosine similarity between expert gate vectors; hidden-state geometry	Suppress redundant co-activation or explain routing by representation similarity
Local GP aggregation	Sparse precision matrix over expert predictions	Keep the most strongly connected experts
Dexterous grasping	Shape-based clustering and point-cloud gating	Select the grasp expert predicted to succeed on the object geometry

A recurrent motivation is that expert systems often fail not because they lack parameters, but because they activate the wrong subset of experts. In MoEs this appears as redundant co-activation of functionally similar experts; in local GP ensembles it appears as the inclusion of weak experts that increase cost and can harm prediction quality; in robotic grasping it appears as the use of a grasp controller mismatched to object shape (Zheng et al., 15 Oct 2025, Jalali et al., 2021, Xu et al., 28 Sep 2025).

The literature also draws an important boundary around the term. GatePro, for example, is explicitly described as a routing-time expert selection optimization method for sparse MoE models, not as a conventional geometry-based expert selection method in the sense of explicitly optimizing distances, manifolds, or geometric assignment constraints. Its relation to GES is indirect: it uses a geometry-like similarity signal in gating space to identify redundant experts and then applies localized competition (Zheng et al., 15 Oct 2025).

2. Geometric signals and mathematical formulations

The defining operation in GES is the conversion of a structured signal into an expert-selection rule. The signal varies by domain.

In GatePro, expert proximity is measured by cosine similarity between rows of the router weight matrix:

$S_{ij} \coloneqq \frac{\langle\mathbf{w}_{g,i}, \mathbf{w}_{g,j}\rangle}{\|\mathbf{w}_{g,i}\| \cdot \|\mathbf{w}_{g,j}\|}.$

For each expert $i$ , the method identifies the most similar counterpart

$j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$

and then imposes pairwise winner-take-all competition on the current token, penalizing the losing expert’s logit by a fixed positive constant, typically $\lambda = 10^{-4}$ (Zheng et al., 15 Oct 2025).

In the hidden-state geometry view of MoE routing, the router is a linear map

$g_t^\ell = P^\ell h_t^\ell,$

so routing similarity is determined by similarity of hidden states along directions that the router retains. The paper formalizes this with a projection-based bound:

$\|P h_i - P h_j\|_2 \le \|P\Pi_r\|_2 \,\| \Pi_r(h_i-h_j)\|_2 + \|P(I-\Pi_r)(h_i-h_j)\|_2,$

where $\Pi_r$ projects onto the top- $r$ principal subspace of the hidden states (Wang et al., 10 Apr 2026).

In Gaussian Experts Selection using graphical models, the relevant structure is an undirected dependency graph over local expert predictions. A sparse precision matrix is estimated by Graphical Lasso:

$\widehat{\Omega}_{\lambda} = \arg\max_{\Omega} \log |\Omega| - \operatorname{trace}(S \Omega) - \lambda \|\Omega\|_1,$

and expert importance is then defined by

$\mathcal{I}_i=\sum_{j=1, j\neq i}^M |\widehat{\Omega}_{\lambda,ij}|.$

Experts are sorted by $i$ 0, and only the top fraction $i$ 1 is retained (Jalali et al., 2021).

In GES-UniGrasp, the geometry signal is the object point cloud. Objects are first grouped by geometry-similar clustering, and a gating network then predicts a softmax distribution over expert success rates from the point cloud. The gating network is trained with KL divergence,

$i$ 2

so the selected expert is the one with the highest predicted score for the current object geometry (Xu et al., 28 Sep 2025).

These formulations differ substantially, but they share a common structure: a geometry-like statistic is computed, converted into a local notion of relevance or conflict, and then used to prune, suppress, or select experts.

3. Similarity-guided routing in sparse Mixture-of-Experts models

In sparse MoEs, standard routing selects the top- $i$ 3 experts from learned router logits. GatePro argues that this routing regime, even when combined with auxiliary balance loss, leaves the core redundancy problem unresolved: experts can be well balanced in token count while still being functionally redundant and co-activated on the same tokens (Zheng et al., 15 Oct 2025).

GatePro therefore targets redundancy directly. Its mechanism is a localized competition overlay on standard token-choice routing. The method computes the cosine-similarity matrix of expert gate vectors, finds each expert’s nearest functional neighbor, suppresses the loser in each similar pair by a small fixed penalty, and only then applies the usual top- $i$ 4 selection. The method is parameter-free, hot-swappable, usable during any training phase, applicable in pretraining and continued training, and deployable without architectural changes (Zheng et al., 15 Oct 2025).

The intended effect is not merely load redistribution. The paper distinguishes three concepts: load balance, specialization, and redundancy. Load balance means experts receive comparable numbers of tokens; specialization means experts develop distinct capabilities; redundancy means experts overlap functionally and are co-activated on similar tokens. GatePro is presented as addressing the second and third items rather than directly optimizing the first (Zheng et al., 15 Oct 2025).

Its diagnostics are explicitly geometric. The appendix defines average cosine similarity, average angle, and spectral entropy over the expert similarity structure. GatePro is reported to produce lower cosine similarity, larger angles, and higher spectral entropy, indicating a more diverse and less redundant expert set. Zero-token-count tracking further shows that unused experts become active faster, and the effect is stronger in deeper layers and with $i$ 5 experts (Zheng et al., 15 Oct 2025).

The reported benchmark gains are modest but systematic. On Seed-MoE-0.7B/7B and 1.3B/13B, GatePro improves results across MMLU-Pro, MMLU, BBH, HellaSwag, GSM8K, and MBPP. On CT-stage evaluation, overall scores improve from $i$ 6 to $i$ 7 on $i$ 8B/ $i$ 9B and from $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 0 to $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 1 on $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 2B/ $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 3B. On OLMoE-1B/7B, overall performance improves from $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 4 to $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 5. The paper also reports that GatePro and balance loss are complementary, and that partial early use can leave a persistent training legacy even after the method is turned off (Zheng et al., 15 Oct 2025).

In relation to GES proper, GatePro occupies an intermediate position. It does not formulate expert routing as a geometric optimization problem over a manifold, Voronoi partition, or explicit clustering objective. It does, however, use a cosine-similarity geometry in gating-weight space to decide which experts should compete, making it a similarity-guided competitive routing mechanism rather than a full standalone geometry-based selector (Zheng et al., 15 Oct 2025).

4. Hidden-state geometry and the interpretation of expert specialization

A stronger geometrical claim appears in "The Myth of Expert Specialization in MoEs: Why Routing Reflects Geometry, Not Necessarily Domain Expertise" (Wang et al., 10 Apr 2026). The central thesis is that expert usage similarity is explained by hidden-state similarity because the router is linear. Under this view, what appears as expert specialization is primarily an emergent property of representation space rather than evidence that the routing module has discovered a human-readable taxonomy of domains.

The paper provides both theoretical and empirical support for this claim. At token level, small hidden-state distance implies small router-logit distance across five pre-trained models, producing a triangle-shaped relation in scatter plots: nearby hidden states reliably route similarly, whereas distant hidden states can still route similarly if the router projects away the relevant differences. At sequence level, cosine similarity of pooled hidden states correlates strongly with similarity of expert-frequency distributions, explaining why routing can appear domain-specific at a coarse scale (Wang et al., 10 Apr 2026).

The same work also analyzes auxiliary load-balancing loss. Under a correlated-data model $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 6, and a small-logit approximation to softmax, the balancing loss is approximated by

$j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 7

This implies that any near-minimizer must satisfy $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 8 up to the softmax’s invariance to constant shifts, so shared directions become routing-insignificant. The paper argues that this offers a theoretical explanation for specialization collapse under small-batch load balancing (Wang et al., 10 Apr 2026).

Several empirical findings challenge a naive interpretation of experts as stable semantic modules. Different models answering the same question show only about $j^*(i) \coloneqq \arg\max_{j \neq i} S_{ij},$ 9 overlap in top experts, which is comparable to the overlap between the same model on different questions. Prompt-level routing does not reliably predict rollout-level routing. In deeper layers, semantically unrelated inputs can converge to near-identical expert activation during prefilling, even though those pruned experts become important again during generation. Auxiliary-loss-trained models show stronger depthwise correlation and router suppression of shared directions than auxiliary-loss-free models such as Moonlight and Trinity Mini (Wang et al., 10 Apr 2026).

A common misconception is therefore that expert specialization in MoEs is inherently interpretable as domain expertise. The evidence instead supports a more restricted statement: routing reflects hidden-state geometry, and any apparent specialization is mediated by that geometry. This does not mean specialization is absent; it means that its interpretation is constrained by the structure of the representation space (Wang et al., 10 Apr 2026).

5. Graphical-model selection of Gaussian experts

"Gaussian Experts Selection using Graphical Models" addresses a different expert-selection problem: local Gaussian-process approximations for large datasets (Jalali et al., 2021). The conventional tradeoff is between conditionally independent aggregation, which is fast but can provide poor uncertainty quantification, and dependent-expert aggregation, which is statistically stronger but computationally expensive because it requires inversion of an $\lambda = 10^{-4}$ 0 covariance matrix at every test point.

The paper’s solution is a theory-guided selection step that prunes weak experts before dependent aggregation. Local expert predictions

$\lambda = 10^{-4}$ 1

are modeled as jointly Gaussian, with conditional dependencies encoded by a sparse precision matrix. In a Gaussian graphical model, $\lambda = 10^{-4}$ 2 means experts $\lambda = 10^{-4}$ 3 and $\lambda = 10^{-4}$ 4 are conditionally independent given the rest, while nonzero entries define the dependency graph. The magnitude of $\lambda = 10^{-4}$ 5 measures interaction strength, and the row-wise sum of absolute off-diagonal entries becomes the expert-importance score $\lambda = 10^{-4}$ 6 (Jalali et al., 2021).

After pruning, aggregation is performed only on the selected subset:

$\lambda = 10^{-4}$ 7

The method preserves the dependent-aggregation form of NPAE while reducing cost. The full NPAE complexity is $\lambda = 10^{-4}$ 8, whereas the selected-expert version is summarized as

$\lambda = 10^{-4}$ 9

The paper further states that the selected-expert estimator is Gaussian, consistent, and asymptotically equivalent to the full NPAE predictor (Jalali et al., 2021).

This formulation is geometry-based in a graph-theoretic sense rather than an Euclidean one. The selection rule does not ask which expert is nearest in input space; it asks which experts are central in the dependency structure induced by covariance and precision. The practical effect is to remove weak or nearly irrelevant experts while preserving calibrated uncertainty quantification. The paper reports that NPAE $g_t^\ell = P^\ell h_t^\ell,$ 0, the selected-expert version, often matches or improves on NPAE, that even using only $g_t^\ell = P^\ell h_t^\ell,$ 1 of experts can approximate NPAE well, and that CI-based baselines improve when the same pruning principle is applied (Jalali et al., 2021).

6. Geometry-based expert selection in dexterous grasping

The most explicit use of the term appears in "GES-UniGrasp: A Two-Stage Dexterous Grasping Strategy With Geometry-Based Expert Selection" (Xu et al., 28 Sep 2025). Here GES is motivated by the claim that dexterous grasping policies are highly sensitive to object shape, size, and local affordances, and that a single monolithic policy tends to collapse toward a narrow grasp style. The proposed remedy is to partition objects into geometry-similar groups, train specialized experts on those groups, and then use a geometry-based gating network to select the expert most likely to succeed at test time.

The framework is built on ContactGrasp, a dataset with $g_t^\ell = P^\ell h_t^\ell,$ 2 objects in $g_t^\ell = P^\ell h_t^\ell,$ 3 categories, split into $g_t^\ell = P^\ell h_t^\ell,$ 4 training objects, $g_t^\ell = P^\ell h_t^\ell,$ 5 seen-category but unseen-instance objects, and $g_t^\ell = P^\ell h_t^\ell,$ 6 unseen-category objects. The dataset is designed to account for task-relevant wrist orientation and thumb-index pinching coordination. These design choices matter because the expert policies are not trained from arbitrary initial conditions; they are trained from contact-guided and retargeted grasp poses intended to be human-like and physically plausible (Xu et al., 28 Sep 2025).

The GES mechanism has two stages. First, a contact-guided synthesis module produces a grasp pose and a pre-grasp pose from the object point cloud. The method uses PCA to choose horizontal versus vertical grasping, canonical alignment rules that distinguish cylindrical from non-cylindrical objects, contact computation by slicing and extremum analysis, retargeting under joint limits, and collision-aware refinement with a PD controller. The control law is

$g_t^\ell = P^\ell h_t^\ell,$ 7

and a final transformation shifts the grasp by $g_t^\ell = P^\ell h_t^\ell,$ 8 m along the approach direction to create the pre-grasp pose $g_t^\ell = P^\ell h_t^\ell,$ 9 (Xu et al., 28 Sep 2025).

Second, a geometry-based multi-policy execution module selects and runs one expert. Objects are grouped by K-means and category-based clustering into geometry-similar groups, and each expert is trained by a three-stage curriculum: intra-cluster central training, intra-cluster expansion, and inter-cluster generalization. A hard-case expert is additionally trained on objects that are difficult for the top two experts, especially those lacking pinch surfaces or exhibiting large centroid shifts (Xu et al., 28 Sep 2025).

The specialization analysis is explicitly geometric. Expert 1 prefers small, flat objects; Expert 2 prefers medium-sized objects and objects with slight initial rotations; Expert 3 prefers larger, box-shaped objects and objects requiring stronger pinch force. The paper further reports more balanced expert usage, less weight oscillation, less redundancy, and more effective expert allocation than ResDex. In this domain, GES is not merely an analysis lens; it is the central mechanism that links object geometry, policy specialization, and test-time gating (Xu et al., 28 Sep 2025).

The acronym “GES” is overloaded across the literature, and not every expert-selection method is geometry-based.

In control and reinforcement learning, "Expert Selection in High-Dimensional Markov Decision Processes" formulates online expert selection as a multi-armed bandit problem over a fixed set of candidate policies (Rubies-Royo et al., 2020). The controller executes one expert for an episode of length $r$ 5, measures episode-average reward, and uses a modified UCB rule

$r$ 6

to balance exploration and exploitation. The method provides logarithmic regret guarantees under irreducibility and aperiodicity assumptions, and its high-dimensional claim arises from the dependence of the bias bound on the second-largest eigenvalue of the induced Markov matrix rather than on observation-space dimension. This is expert selection, but it is reward-driven online switching rather than GES (Rubies-Royo et al., 2020).

In causal discovery, “GES” refers to Greedy Equivalence Search, not to geometry-based expert selection. "Reframed GES with a Neural Conditional Dependence Measure" revisits score-based structure learning of a Markov equivalence class and replaces the conventional graph score with a conditional dependence statistic $r$ 7, preserving the forward and backward equivalence-search structure (Shen et al., 2022). Similarly, "Effects of Nonparanormal Transform on PC and GES Search Accuracies" studies when a marginal Gaussianization transform helps causal graph search procedures such as GES-BIC and PC-GES under moderate non-Gaussianity and moderate non-linearity (Ramsey, 2015). These works are unrelated to expert selection despite sharing the acronym.

A second source of confusion concerns what “geometry-based” means. In robotics, it means explicit clustering and gating on object shape (Xu et al., 28 Sep 2025). In local GP aggregation, it means structural centrality in a graphical model over experts (Jalali et al., 2021). In MoEs, it may refer either to geometry-like similarity in gating-weight space, as in GatePro, or to the more general claim that routing behavior is determined by hidden-state geometry (Zheng et al., 15 Oct 2025, Wang et al., 10 Apr 2026). The term therefore spans Euclidean geometry, graph geometry, and representation geometry.

A final misconception is that balanced expert utilization is equivalent to meaningful expert diversity. GatePro explicitly argues otherwise: balanced token counts are a statistical property of routing load, whereas specialization diversity is functional, and experts can remain redundant even under good load balance (Zheng et al., 15 Oct 2025). Relatedly, the hidden-state geometry analysis argues that expert specialization should not be equated with stable human-interpretable domain expertise; much of what appears as specialization may instead reflect the geometry of representation space (Wang et al., 10 Apr 2026).

Taken together, these distinctions indicate that GES is best treated as a family resemblance concept. Its instances differ in objective, representation, and optimization strategy, but they converge on a common premise: expert systems improve when selection is informed by the structured geometry of experts, inputs, or their induced interactions.