Probabilistic Landmarks Overview

• Probabilistic landmarks are representations that model uncertainty by replacing fixed points with probability distributions across various domains.
• They are applied in fields like shape analysis, SLAM, localization, and navigation, using stochastic processes and Bayesian inference to quantify uncertainty.
• This research program unifies methods that employ diffusion processes, posterior densities, and decision-theoretic planning to enhance model reliability.

Searching arXiv for recent and foundational papers on probabilistic landmarks across geometry, shape analysis, SLAM, navigation, and vision. Probabilistic landmarks are landmark representations in which uncertainty is modeled explicitly rather than suppressed into a single deterministic point, label, or subgoal. In current research usage, the term spans several technically distinct constructions: stochastic landmark manifolds in shape analysis, posterior distributions over landmark locations and counts, landmark persistence and data-association uncertainty in SLAM, stochastic-geometric models of anonymous landmarks for localization, and landmark selection or ordering schemes for navigation and planning. This suggests that “probabilistic landmarks” is best read as an umbrella term whose precise meaning is field-dependent, but whose common thread is the replacement of fixed landmark assertions by distributions, random sets, belief states, or uncertainty-aware decision criteria (Habermann et al., 1 Jun 2026, Strait et al., 2017, Kumar et al., 2020).

1. Scope and meanings of the term

In the recent literature, the phrase does not denote a single canonical model. Instead, it refers to several families of methods that place probability on different objects associated with landmarks.

Research line	What is probabilistic	Representative papers
Shape-space geometry	Brownian landmark dynamics, collision/explosion, stochastic completeness	(Habermann et al., 2023, Habermann et al., 1 Jun 2026)
Statistical landmark inference	Landmark locations, landmark number, visibility, anisotropic covariance	(Strait et al., 2017, Kumar et al., 2020)
Localization, SLAM, mapping	Persistence states, data associations, candidate landmark sets, semantic cell beliefs	(Bateman et al., 2020, Zhang et al., 2023, Hu et al., 2024, Hu et al., 30 Jan 2025, Paz et al., 2020, Bhamidipati et al., 2021)
Planning and navigation	Selection probabilities, stochastic subgoal landmarks, multiverse landmark utility, probabilistic ordering graphs	(Mousavi et al., 2019, Chan et al., 15 Aug 2025, Chen et al., 31 May 2026, Pérez-Corral et al., 21 Sep 2025)

A recurring source of confusion is that some works are merely landmark-based, while the probabilistic component belongs elsewhere. In landmark-guided probabilistic roadmap queries, for example, the roadmap is probabilistic because it is built by random sampling, whereas the landmarks are selected roadmap vertices used for an admissible heuristic; they are not themselves probabilistic state variables (Paden et al., 2017). A similar contrast appears in hierarchical goal-reaching with replay-buffer landmarks: the background MDP is stochastic, but the landmark machinery is based on farthest-point sampling and deterministic shortest-path routing rather than explicit probabilistic landmark inference (Huang et al., 2019).

2. Stochastic landmark geometry in shape analysis

One major meaning of probabilistic landmarks arises in statistical shape analysis and the large deformation diffeomorphic metric mapping framework. Here the landmark space is the ordered collision-free configuration space

$\Land=\{x=(x_1,\dots,x_n): x_i\in \mathbb R^d,\ x_i\neq x_j \text{ for } i\neq j\},$

equipped with a kernel-induced Riemannian metric whose cometric satisfies

$g^{ij}(x)=K(x_i,x_j),$

and, under translation and rotation invariance,

$K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$

The probabilistic object is intrinsic Brownian motion on $(\Land,g)$, together with associated diffusion processes used for stochastic shape evolution, diffusion bridges, and Brownian priors (Habermann et al., 1 Jun 2026).

The central question is whether these diffusion processes exist for all times or can explode by hitting the collision set. For configurations of exactly two landmarks, the inter-landmark distance reduces the problem to a one-dimensional diffusion, and the sharp threshold is governed by the near-zero asymptotics

$k(0)-k(r)=Dr^\gamma+o(r^\gamma).$

In that case, Brownian completeness holds if $\gamma\ge 2$ and fails if $\gamma<2$; low-order Sobolev kernels permit finite-time collision with positive probability, whereas higher-order Sobolev and Gaussian kernels yield long-time existence (Habermann et al., 2023). This established that kernel regularity is not merely a smoothing choice but a probabilistic well-posedness condition.

The arbitrary-$n$ case was subsequently addressed through stochastic completeness results for landmark spaces of any cardinality. The key theorem requires a near-diagonal bound

$k(0)-k(r)\le -c r^2\log r$

for small $r$, together with a high-frequency Fourier lower bound

$g^{ij}(x)=K(x_i,x_j),$0

for sufficiently large $g^{ij}(x)=K(x_i,x_j),$1. Under these assumptions, $g^{ij}(x)=K(x_i,x_j),$2 is stochastically complete for all $g^{ij}(x)=K(x_i,x_j),$3 (Habermann et al., 1 Jun 2026). The proof combines Grigor’yan’s volume growth criterion with quantitative control of geodesic balls: Euclidean size grows at most linearly in radius, pairwise separations inside a geodesic ball decay only exponentially in $g^{ij}(x)=K(x_i,x_j),$4, and the minimal eigenvalue of the cometric is controlled from below via the Fourier transform of the kernel. For Matérn kernels induced by

$g^{ij}(x)=K(x_i,x_j),$5

this yields the threshold

$g^{ij}(x)=K(x_i,x_j),$6

These results matter directly for probabilistic landmark models because stochastic completeness is exactly the non-explosion property for intrinsic Brownian landmark dynamics. In this setting, “probabilistic landmarks” does not refer to uncertain landmark annotations; it refers to stochastic processes evolving on a landmark manifold whose geometry degenerates near collisions. A practical implication is that Brownian simulation, diffusion-based inference, and heat-kernel constructions on landmark spaces are justified only for kernel classes whose small-scale and Fourier behavior exclude finite-time explosion (Habermann et al., 1 Jun 2026).

3. Bayesian inference of landmark locations, counts, and visibility

A second, more classical meaning of probabilistic landmarks appears when landmark locations themselves are treated as latent random variables. In elastic shape analysis of curves, landmarks are inferred as domain positions

$g^{ij}(x)=K(x_i,x_j),$7

rather than as pre-annotated Euclidean points. The model in "Automatic Detection and Uncertainty Quantification of Landmarks on Elastic Curves" represents each observed curve by its square-root velocity function and defines a piecewise-linear reconstruction through candidate landmarks. The discrepancy is modeled in SRVF space by

$g^{ij}(x)=K(x_i,x_j),$8

with white-noise covariance after discretization (Strait et al., 2017).

The main inferential device is a reparameterization from ordered landmarks to simplex-valued spacings. For open curves,

$g^{ij}(x)=K(x_i,x_j),$9

so that $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$0 and $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$1. A symmetric Dirichlet prior on $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$2 and a Gamma prior on the precision $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$3 induce a posterior over landmark configurations. The paper also treats the landmark number $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$4 itself as random, using a shifted Poisson prior and reversible-jump MCMC. In this formulation, probabilistic landmarks are posterior objects: uncertainty is expressed through marginal densities, credible intervals, multimodality, posterior dependence among landmarks, and a posterior histogram over $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$5 (Strait et al., 2017).

Face alignment extends the same idea to per-landmark location uncertainty and landmark visibility. LUVLi formulates each facial landmark as a mixed random variable: a binary visibility variable $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$6 and a continuous location conditional on visibility. The predictive model factorizes as

$K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$7

where $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$8 is Bernoulli and $K(x_i,x_j)=k(\|x_i-x_j\|)I_d.$9 is either a 2D Gaussian or a 2D Laplacian with full covariance

$(\Land,g)$0

The negative log-likelihood combines visibility classification and conditional location likelihood, so self-occluded landmarks contribute only visibility loss, while unoccluded and externally occluded landmarks contribute both (Kumar et al., 2020).

What is notable here is that the uncertainty is anisotropic and landmark-specific. Mean location is extracted from a heatmap by a differentiable spatial mean, but covariance is regressed separately through a Cholesky factorization $(\Land,g)$1, and visibility is estimated by an additional Bernoulli head. The resulting model outputs, for each landmark, a point estimate $(\Land,g)$2, a full covariance $(\Land,g)$3, and a visibility probability $(\Land,g)$4. This is a stronger probabilistic object than a scalar confidence score, because it encodes directional uncertainty and the possibility that no meaningful location exists (Kumar et al., 2020).

Taken together, these works define probabilistic landmarks as latent variables over which one performs posterior inference. Depending on the application, the random quantity may be the landmark’s position along a curve, the number of landmarks required for reconstruction, the existence of a labeled location, or the full covariance ellipse around a predicted point.

4. Localization, SLAM, and mapping with uncertain landmarks

In robot localization and SLAM, probabilistic landmarks often denote uncertainty not about whether landmarks are mathematically necessary, but about which landmark was observed, whether a landmark still exists, how reliable a landmark is, or whether an indistinguishable landmark configuration is sufficiently unique for localization.

Data-association-free landmark SLAM makes the uncertainty over landmark identity explicit. The robot must jointly estimate trajectory $(\Land,g)$5, landmark positions $(\Land,g)$6, unknown correspondence variables $(\Land,g)$7, and even the number of landmarks $(\Land,g)$8. For fixed $(\Land,g)$9, the inner optimization is

$k(0)-k(r)=Dr^\gamma+o(r^\gamma).$0

and an outer penalty $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$1 regularizes the unknown landmark count (Zhang et al., 2023). For fixed trajectory, the problem reduces to $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$2-means on projected observations, so landmarks become latent cluster centers under hard assignment. This is probabilistic in the sense of latent uncertainty under Gaussian noise, but the solution is MAP-like rather than a full posterior.

Long-term mapping introduces another landmark uncertainty: persistence. In online probabilistic clique change detection for 3D landmark-based maps, landmarks are grouped into cliques that are presumed to arise from a common geometric object or structure, and a Bayesian filter estimates their persistence jointly over time. The latent quantity is a binary persistence state $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$3 per landmark, aggregated into clique states $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$4, with inference operating on posteriors of the form

$k(0)-k(r)=Dr^\gamma+o(r^\gamma).$5

This allows dynamic and semi-static objects to be removed from nominally static maps using clique-level dependence rather than independent per-landmark scoring (Bateman et al., 2020). The probabilistic landmark here is a map entity augmented with a posterior probability of still belonging to the persistent world.

A different localization regime arises when landmarks are visible but not uniquely identifiable. In a stochastic-geometric treatment, landmark locations are modeled as a Poisson point process $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$6, and localizability is defined through the probability that noisy measurement sets at two locations are disjoint: $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$7 For range-only and snapshot-like measurement models, expressions and bounds for $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$8 are derived as functions of landmark intensity, visibility radius, and noise. A key conclusion is that localizability approaches one as landmark intensity tends to infinity, so anonymous landmark constellations can become asymptotically unique even when individual landmarks are indistinguishable (Hu et al., 2024). A related marked-PPP model, where marks denote visual types, shows that in the noise-free two-dimensional case three range measurements are sufficient almost surely to identify the correct landmark combination, while the noisy case admits an explicit localizability probability based on true-positive and false-positive retention of candidate combinations (Hu et al., 30 Jan 2025).

Uncertainty can also be semantic rather than geometric. In probabilistic semantic mapping for urban driving, static road landmarks such as roads, sidewalks, crosswalks, and lane marks are represented on a bird’s-eye-view grid, with each cell storing a class distribution rather than a single label. The semantic Bayes update combines image-derived labels, a confusion-matrix observation model, and LiDAR intensity cues: $k(0)-k(r)=Dr^\gamma+o(r^\gamma).$9 Here a “landmark” is a semantic map element rather than a sparse point, and the probability mass is over class membership of map cells (Paz et al., 2020).

A further extension appears in integrity-driven GPS-vision localization, where each GPS satellite and each visual landmark induces a probabilistic zonotope over expected state error. Visual landmarks are especially notable because their own map positions are uncertain, contributing a landmark-position p-Zonotope $\gamma\ge 2$0 to the expected state bound. Landmark selection is then performed by minimizing an integrity-oriented p-Zonotopic cost under subset constraints, so probabilistic landmarks are decision variables whose inclusion or exclusion changes certified localization bounds (Bhamidipati et al., 2021).

5. Landmark selection, subgoals, and planning under uncertainty

A separate line of work treats landmarks as control-relevant intermediates rather than as latent geometric entities. In robot visual navigation over finite horizons, each candidate feature contributes an additive positive-semidefinite term $\gamma\ge 2$1 to the information matrix, so the selected landmark subset $\gamma\ge 2$2 determines

$\gamma\ge 2$3

This makes landmark selection an information-matrix subset problem. A randomized leverage-score scheme assigns each feature a sampling probability

$\gamma\ge 2$4

thereby turning landmarks into probabilistic information sources whose value is measured by expected contribution to trajectory estimation quality (Mousavi et al., 2019).

In stochastic planning, landmarks acquire yet another meaning: conditions that must become true in every successful execution. The probabilistic generalization is defined over successful histories of policies rather than deterministic plans. A condition $\gamma\ge 2$5 is a landmark if, for all policies $\gamma\ge 2$6 and all successful histories $\gamma\ge 2$7, $\gamma\ge 2$8 is true at some time in $\gamma\ge 2$9 (Chan et al., 15 Aug 2025). This preserves natural, necessary, and greedy-necessary ordering notions, and classical landmarks extracted from the all-outcomes determinization remain valid probabilistic landmarks in the stochastic domain. These landmarks are then used inside UCT, where the action criterion mixes current-landmark and final-goal action values: $\gamma<2$0 The landmark is probabilistic not because its coordinates are random, but because necessity is quantified over stochastic histories and because the planner balances subgoal greediness against long-term utility (Chan et al., 15 Aug 2025).

A domain-level planning variant learns a probabilistic lifted ordering graph rather than extracting only task-specific landmark relations. The learned p-LOG stores lifted landmark orderings $\gamma<2$1 with empirical probabilities

$\gamma<2$2

These weights are frequencies across multiple tasks in the same planning domain, not logical guarantees. The resulting representation captures reusable but non-universal landmark structure and can be instantiated for a new problem through forward and backward graph generation from initial and goal states (Pérez-Corral et al., 21 Sep 2025).

Open-vocabulary embodied navigation extends the same idea to semantic scene uncertainty. PSG-Nav builds a 3D probabilistic scene graph with object, group, and room nodes, each object carrying a categorical belief $\gamma<2$3 via vote accumulation. Candidate geometric landmarks are generated from a generalized Voronoi graph and frontiers, then ranked by a combination of spatial and semantic information gain,

$\gamma<2$4

and by a multiverse decision score averaged across sampled deterministic scene graphs: $\gamma<2$5 The selected landmark maximizes $\gamma<2$6 (Chen et al., 31 May 2026). In this formulation, a landmark is probabilistic because its semantic compatibility with the goal depends on uncertain object, group, and room interpretations, not because its location is itself random.

6. Recurring principles, misconceptions, and limits

Across these fields, probabilistic landmarks differ in ontology but share a small set of recurring principles. First, the random quantity may live at very different levels: a point on a curve, a landmark count, a visibility variable, a persistence state, a map-cell class, a candidate correspondence, a subgoal utility, or the long-time existence of a diffusion on landmark space. Second, uncertainty is often structured rather than independent: clique-level persistence ties landmarks to common objects (Bateman et al., 2020), hierarchical scene graphs tie objects to groups and rooms (Chen et al., 31 May 2026), and kernel geometry couples pairwise landmark distances through the cometric (Habermann et al., 1 Jun 2026). Third, many methods are explicitly decision-theoretic: posterior uncertainty is not merely reported but used to select landmarks, prune associations, reject false positives, or bias planners (Mousavi et al., 2019, Bhamidipati et al., 2021, Chan et al., 15 Aug 2025).

A common misconception is that all probabilistic-landmark methods are Bayesian coordinate regressors. That is only one subcase. The elastic-curve and LUVLi models indeed place probability on landmark locations and counts (Strait et al., 2017, Kumar et al., 2020), but other strands place it on persistence, association, semantic identity, or necessity relations (Zhang et al., 2023, Hu et al., 30 Jan 2025, Pérez-Corral et al., 21 Sep 2025). Another misconception is that any landmark method inside a stochastic system is automatically a probabilistic landmark model. The PRM and replay-buffer landmark works show the opposite: landmarks may remain deterministic anchors even when the overall framework is probabilistic or based on an MDP (Paden et al., 2017, Huang et al., 2019).

The limits are equally field-specific. Stochastic completeness conditions for general kernels are sufficient rather than necessary beyond the covered classes (Habermann et al., 1 Jun 2026). Bayesian landmark detection on curves depends on a homogeneous population assumption and can face multimodality and sharply peaked likelihoods (Strait et al., 2017). LUVLi models per-landmark uncertainty with a single Gaussian or Laplacian and therefore cannot represent multimodal ambiguity (Kumar et al., 2020). Data-association-free SLAM resolves uncertainty through hard assignments and model-order penalties rather than full posterior marginalization (Zhang et al., 2023). Stochastic-geometric localization relies on homogeneous PPP assumptions and bounded-noise abstractions (Hu et al., 2024, Hu et al., 30 Jan 2025). Planning methods based on probabilistic landmarks can improve finite-budget decision making, but greedy landmark pursuit is not complete and may conflict with global safety (Chan et al., 15 Aug 2025).

Taken together, these works suggest that probabilistic landmarks are best regarded not as a single model class but as a research program: landmarks are no longer treated as immutable discrete facts, but as uncertainty-bearing geometric, semantic, statistical, or decision-theoretic entities whose probabilistic structure is essential to inference, control, or long-time well-posedness.