Uni-Mapper: Unified Graph and Map Construction

• Uni-Mapper is a polysemous term that unifies topological data analysis and robotic mapping through distinct, context‐driven frameworks.
• In TDA, it refers either to a univariate Mapper built from scalar filters or to a probabilistic, uncertainty-aware mapper that learns interval covers automatically.
• In robotics, it encompasses both a standardized map-handling workflow integrating spatial primitives and a dynamic-aware LiDAR map-merging framework enhancing multi-modal localization.

Uni-Mapper is a polysemous research term rather than a single canonical method. In topological data analysis, it denotes either the univariate Mapper $M(f,U)$ built from a single scalar filter, or a unified, uncertainty-aware, automatically optimized Mapper framework based on implicit intervals and stochastic optimization. In robotics, it denotes either a shorthand for a standardized map-handling workflow centered on location, objects, occupancy, and topology, or a dynamic-aware 3D point-cloud map-merging framework for multi-modal LiDARs. The common thread across these usages is unification: either of cover construction and graph summarization in Mapper, or of heterogeneous spatial representations and sensing modalities in robotic mapping (Zhou et al., 2021, Tao et al., 2024, Heselden et al., 2024, Kang et al., 28 Jul 2025, Carrière et al., 2015).

1. Terminological scope

The term has distinct meanings in separate literatures, and its interpretation depends on context. In the TDA literature, one usage is essentially descriptive: Uni-Mapper means the univariate specialization of Mapper. A second TDA usage introduces Uni-Mapper as a probabilistic, learnable Mapper pipeline. In robotics, one usage is explicitly informal: a summary employs Uni-Mapper as a shorthand for a unified map-handling approach even though the paper’s formal title is different. A later robotics paper uses Uni-Mapper as the proper name of a LiDAR map-merging framework (Zhou et al., 2021, Tao et al., 2024, Heselden et al., 2024, Kang et al., 28 Jul 2025).

Context	Meaning of “Uni-Mapper”	Reference
Stitch Fix / TDA	Univariate Mapper built from one scalar filter	(Zhou et al., 2021)
Probabilistic Mapper / TDA	Unified, uncertainty-aware, automatically optimized Mapper	(Tao et al., 2024)
Unified map handling / robotics	Shorthand for an information-centric map-handling approach; not the formal name	(Heselden et al., 2024)
Multi-modal LiDAR mapping / robotics	Proper name of a dynamic-aware 3D point-cloud map-merging framework	(Kang et al., 28 Jul 2025)

A recurring misconception is to treat Uni-Mapper as a single algorithmic lineage. The record instead shows a term reused across at least two major domains: topological summarization of data and robotic map representation or merging.

2. Uni-Mapper as univariate Mapper and stitched Mapper composition

In "Stitch Fix for Mapper and Topological Gains" (Zhou et al., 2021), Uni-Mapper denotes the univariate Mapper construction. Let $X$ be a topological space and $f:X\to\mathbb{R}$ a continuous filter. If $U=\{U_1,\dots,U_n\}$ is a cover of $f(X)$ by intervals, each inverse image $f^{-1}(U_i)$ is decomposed into path-connected components $u_{i1},\dots,u_{ik_i}$, and the Mapper is defined as

$M(f,U):=Nrv(f^*(U)).$

In practical point-cloud settings, the construction is parameterized by a resolution $n$, an overlap $p$, and a clustering algorithm such as DBSCAN used to approximate the connected components of each pullback.

The same paper develops a stitching procedure that composes two univariate mappers into a bivariate mapper. Given $X$0 with covers $X$1 and $X$2, the composed cover is built from the path-connected components of nonempty intersections between pullback cover elements of $X$3 and $X$4, and the composed mapper is

$X$5

Under the stated assumptions that $X$6 and $X$7 are continuous and that the sets $X$8, $X$9, and $f:X\to\mathbb{R}$0 are simply connected, the stitched construction equals the standard bivariate mapper:

$f:X\to\mathbb{R}$1

A central contribution of this usage is the notion of topological gains, which quantify the information added by stitching a second filter onto a first. The paper defines localized homological difference (LHD), local relative Euler characteristic (LREC), and localized entropy differences based on distance-matrix entropy and adjacency entropy, denoted $f:X\to\mathbb{R}$2 and $f:X\to\mathbb{R}$3. These are computed per interval of the first filter, allowing a local comparison between the univariate and stitched constructions. For a cylinder with filters $f:X\to\mathbb{R}$4 and $f:X\to\mathbb{R}$5, $f:X\to\mathbb{R}$6, and $f:X\to\mathbb{R}$7, the paper reports $f:X\to\mathbb{R}$8 and $f:X\to\mathbb{R}$9, indicating that stitching $U=\{U_1,\dots,U_n\}$0 onto $U=\{U_1,\dots,U_n\}$1 yields a $U=\{U_1,\dots,U_n\}$2-cycle in every interval of $U=\{U_1,\dots,U_n\}$3 while the reverse does not add local $U=\{U_1,\dots,U_n\}$4-cycles. For a sphere with filters $U=\{U_1,\dots,U_n\}$5 and $U=\{U_1,\dots,U_n\}$6, $U=\{U_1,\dots,U_n\}$7 and $U=\{U_1,\dots,U_n\}$8, the boundary-subgraph $U=\{U_1,\dots,U_n\}$9 values are reported as

$f(X)$0

$f(X)$1

and

$f(X)$2

This usage of Uni-Mapper is therefore fundamentally categorical and topological. It refers to a mapper graph generated from one scalar filter, and it serves as the basic unit for a stitched multivariate construction together with interval-local diagnostics of topological information gain.

3. Uni-Mapper as a probabilistic, optimized Mapper framework

"A Mapper Algorithm with implicit intervals and its optimization" introduces Uni-Mapper as a unified, uncertainty-aware, and automatically optimized Mapper framework (Tao et al., 2024). The framework replaces explicit interval covers in filter space by implicit intervals represented through a Gaussian Mixture Model. For data $f(X)$3 and a filter $f(X)$4, with $f(X)$5 typically $f(X)$6, the filtered values are $f(X)$7. Uni-Mapper models these values by

$f(X)$8

with responsibilities

$f(X)$9

Collecting $f^{-1}(U_i)$0 yields a probability matrix $f^{-1}(U_i)$1.

To realize overlap, the framework samples a hidden assignment matrix $f^{-1}(U_i)$2 row-wise from a multinomial distribution with two trials:

$f^{-1}(U_i)$3

This enforces a Mapper-like constraint: each point belongs to at least one and at most two intervals. The pullback of component $f^{-1}(U_i)$4 is then the multiset $f^{-1}(U_i)$5, each $f^{-1}(U_i)$6 is clustered, and the Mapper graph is formed by connecting clusters that share at least one original point.

The framework treats the Mapper graph as a random variable $f^{-1}(U_i)$7 and defines a computationally efficient point estimate, the Mapper graph mode. For each row of $f^{-1}(U_i)$8, if $f^{-1}(U_i)$9, the mode assigns membership only to the most probable component; otherwise it assigns membership to the two most probable components. This avoids Monte Carlo averaging over graphs while retaining an uncertainty-aware construction.

Optimization is end-to-end and topological. The GMM negative log-likelihood is

$u_{i1},\dots,u_{ik_i}$0

and the topological signal is summarized by extended persistence on the graph mode. For a node $u_{i1},\dots,u_{ik_i}$1 with member indices $u_{i1},\dots,u_{ik_i}$2, the node filtration is defined by

$u_{i1},\dots,u_{ik_i}$3

while the persistence summary is the average persistence

$u_{i1},\dots,u_{ik_i}$4

The training objective is

$u_{i1},\dots,u_{ik_i}$5

Mixture weights and variances are reparameterized for SGD by

$u_{i1},\dots,u_{ik_i}$6

The framework is positioned against standard Mapper, F-Mapper, ensemble Mapper, Ball Mapper, D-Mapper, and differentiable Mapper. Its principal claim is not that all Mapper hyperparameters disappear, but that interval locations, widths, and effective overlaps are learned automatically, while $u_{i1},\dots,u_{ik_i}$7 and clustering hyperparameters remain user-specified. It is also explicitly distinguished from UMAP: UMAP produces a low-dimensional embedding, whereas Mapper constructs a graph by covering the image of a filter, pulling back, and clustering.

Empirical evaluation spans synthetic circles, a 3D human dataset, and RNA expression data from the Mount Sinai/JJ Peters VA Medical Center Brain Bank. In the RNA expression study on brain area 36 (PHG, BA36), the dataset contains 215 samples with more than 20k genes and Braak AD stage labels from 0 to 6. The filter is the mean hyperbolic distance computed on 37 gene sites identified by prior work, clustering is agglomerative with threshold 3.13, and the reported setting is $u_{i1},\dots,u_{ik_i}$8, learning rate $u_{i1},\dots,u_{ik_i}$9, and 300 epochs. The optimized graph mode reveals a distinct branch enriched for severe AD, with a $M(f,U):=Nrv(f^*(U)).$0 test yielding $M(f,U):=Nrv(f^*(U)).$1 and 73% of patients in the identified subgroup classified as severe versus 37% in the remainder. The paper also notes limitations: non-differentiability from mode selection and clustering, continued dependence on the choice of $M(f,U):=Nrv(f^*(U)).$2 and clustering method, a fixed rather than learned filter $M(f,U):=Nrv(f^*(U)).$3, and the convenience rather than universality of Gaussian components.

4. Theoretical foundations from the 1-dimensional Mapper

The 2015 paper "Structure and Stability of the 1-Dimensional Mapper" provides the most detailed mathematical framework underlying one-dimensional Mapper constructions (Carrière et al., 2015). For a continuous function $M(f,U):=Nrv(f^*(U)).$4 and a cover $M(f,U):=Nrv(f^*(U)).$5 of its image by intervals, the Mapper is the nerve of the connected pullback cover:

$M(f,U):=Nrv(f^*(U)).$6

where each $M(f,U):=Nrv(f^*(U)).$7 is refined into connected components. When $M(f,U):=Nrv(f^*(U)).$8 is an open, minimal, generic interval cover in $M(f,U):=Nrv(f^*(U)).$9, or gomic, the Mapper is a simple graph. The paper also defines the MultiNerve Mapper $n$0 as the multinerve of the connected pullback cover and shows that there is a canonical projection from the MultiNerve Mapper to the Mapper.

The framework is organized around the Reeb graph

$n$1

where $n$2 iff $n$3 and $n$4 lie in the same connected component of the level set $n$5. For Morse-type functions, the Reeb graph is a multigraph, and its extended persistence diagram decomposes into subdiagrams with a reading dictionary: trunks are $n$6, downward branches are $n$7, upward branches are $n$8, and cycles or holes are $n$9. This yields a precise signature of the Reeb graph.

The core structural theorem states that the MultiNerve Mapper signature is obtained by pruning the Reeb-graph signature with staircase regions determined solely by the interval cover. Ordinary, relative, and extended features are removed when their persistence-diagram points fall inside $p$0, $p$1, and $p$2, respectively. For the Mapper itself, the staircase for cycles is enlarged to $p$3 because the Mapper glues multiple edges with the same endpoints, thereby eliminating cycles that remain in the MultiNerve Mapper. This formalizes the common intuition that Mapper is a pixelized approximation of the Reeb graph whose visible branches and cycles depend on interval endpoints and overlap placement relative to critical values.

The same paper establishes staircase-aware stability. A modified bottleneck distance treats the relevant staircase as a substitute for the diagonal, and for Morse-type $p$4 on $p$5 with gomic $p$6,

$p$7

and similarly for the Mapper. The per-feature interpretation is explicit: the $p$8 distance from a diagram point to the appropriate staircase is a lower bound on the perturbation magnitude needed to eliminate that feature. The paper further extends stability to domain perturbations under Lipschitz, curvature, and convexity-radius assumptions, and it quantifies how modifying the interval cover changes the structure through Hausdorff distances between staircase regions.

Convergence is established in both diagrammatic and metric senses. If the granularity of $p$9 is at most $X$00, then

$X$01

and, once $X$02 is below the smallest vertical distance of any point in $X$03 to the diagonal, the signature coincides. In functional distortion distance, a perturbed telescope construction yields

$X$04

for the MultiNerve Mapper, while the Mapper’s geometric representation satisfies the bound $X$05. The practical guidance extracted in the synthesis is equally important: small overlaps help capture more cycles but make them more sensitive to perturbations, whereas large overlaps suppress spurious cycles and improve stability with respect to holes. This suggests that later Uni-Mapper variants in TDA inherit a parameter-selection problem already made explicit in the one-dimensional theory.

5. Uni-Mapper as unified map handling for robotic systems

"Unified Map Handling for Robotic Systems: Enhancing Interoperability and Efficiency Across Diverse Environments" does not introduce Uni-Mapper as a proper name or acronym; in the summary, Uni-Mapper is used as a shorthand for the paper’s unified map handling approach and associated tooling (Heselden et al., 2024). The core problem is not map construction but reliable handling of heterogeneous maps across environments and platforms. The proposal is an information-centric abstraction combined with a file and template organization that standardizes references rather than forcing all information into a single binary or schema.

The unification is organized around four canonical primitives: location, objects, occupancy, and topology. The environment_template repository is a ROS 2 package that provides a fixed hierarchy grouping files by those four categories, while a root environment.sh exports paths so downstream tools can locate canonical sources regardless of the internal layout. This packaging is intended to reduce risks such as inconsistent map versions, divergent environmental understanding across robots, and hard-to-reproduce deployments.

Supported inputs include KML/KMZ, Datum YAML, satellite or floorplan images, Occupancy Grid files, OctoMap, OpenStreetMap XML, NavGraph YAML, ROS topological maps, OpenRMF site maps, and Gazebo SDF worlds. Demonstrated conversions include extracting paths and lanes from OpenRMF or OpenStreetMap into topological maps, projecting these onto KML, deriving topology from occupancy via skeletonization, slicing simulator worlds into 2D occupancy, and composing KML overlays from occupancy, topology, and fences. The paper distinguishes direct conversions, which re-emit the same information type in a different container, from indirect inference, which generates one primitive from another, such as topology from occupancy, occupancy from simulator objects, or objects from imagery using segmentation or detection. When target formats are incomplete, the approach permits template-based or procedural generation using wave-function collapse, cellular automata, gradient noise, or procedural modeling.

Geospatial consistency is handled through global WGS84 coordinates and a local metric map frame, often ENU-aligned around a chosen origin. Datum YAML defines a GNSS fence and anchor used to tie local map frames to global coordinates. Transform composition follows standard homogeneous-transform conventions in $X$06, including

$X$07

If live sensor data are integrated into occupancy layers, the paper notes that a standard log-odds update may be used, but it does not prescribe a specific mapping backend. Optional registration or merging can be performed upstream, for example with ICP or pose-graph optimization, after which the conversions normalize outputs.

The tooling consists principally of the open-source ROS 2 package environment_common and the organizational scaffold environment_template. Public repositories are provided under LCAS. Datasets span agricultural environments at Riseholme Park Farm, urban roadway collections derived from OSM for cities in Asia, Europe, and North America, and indoor office, warehouse, airport, and hospital-like layouts, some of them procedurally generated. The contribution is explicitly organizational and procedural rather than a novel mapping algorithm: no timing, CPU, memory, or throughput benchmarks are reported; automated conflict resolution and multi-source uncertainty fusion are not part of the current tooling; location cannot be inferred without external input such as Datum; and dynamic map updates and multi-robot live synchronization are outside scope.

6. Uni-Mapper as a dynamic-aware LiDAR map-merging framework

"Uni-Mapper: Unified Mapping Framework for Multi-modal LiDARs in Complex and Dynamic Environments" uses Uni-Mapper as the formal name of a dynamic-aware 3D point-cloud map-merging framework for multi-session and multi-robot operation with heterogeneous LiDARs (Kang et al., 28 Jul 2025). Its design targets modality gaps caused by differing scan patterns, fields of view, angular resolution, vertical layering, and nonrepetitive sampling, as well as inconsistencies introduced by dynamic objects such as pedestrians, cars, and buses. The framework integrates three modules: online dynamic object removal, dynamic-aware loop closure, and multi-map merging.

Dynamic object removal is based on a voxel-wise free-space hash map built in a coarse-to-fine manner. Keyframes are accumulated and voxelized with a coarse size $X$08, local PCA is used to identify planar structure, and ground voxels are selected using an angle threshold $X$09 together with a no-points-below-plane condition. Candidate voxels near ground are further subdivided at leaf size $X$10. For each current fine candidate voxel, a sliding-window binary Bayes update is applied to occupancy and free-space evidence, and if the posterior probability exceeds $X$11, non-ground points in that voxel are classified as dynamic and removed. The same voxel structure is shared by the descriptor pipeline, so only static-preserved points contribute to loop-closure features.

The loop-closure module augments the STD descriptor into DynaSTD by extracting triangles only from static points. Each triangle encodes three sorted side lengths and dot products of projection normals, and a hash table stores these attributes for retrieval. Loop candidates are verified by a plane-overlap ratio, and relative pose is estimated in closed form from matched triangle vertices using SVD rather than ICP. For inter-map matching, triangles are recomputed in ego-centric keyframe coordinates to reduce voxel discretization bias across map frames.

Map merging proceeds in two stages. First, each session undergoes intra-session pose-graph optimization with odometry and DynaSTD loop closures, using the robust loss

$X$12

Second, sessions are merged through centralized inter-map pose-graph optimization with anchor nodes. A central map $X$13 is selected, query maps $X$14 are aligned to it through inter-map loop detections, and an anchor-node factor couples the sessions while mitigating residual intra-session drift. After this initial alignment, a radius-based loop search adds dense local inter-map constraints, and G-ICP refines the transformations. The merged map is then maintained with per-voxel kd-trees for scalable nearest-neighbor queries.

The reported implementation is in C++ on ROS Noetic and Ubuntu 20.04, tested on an Intel i7-12700KF with 64 GB RAM. The dynamic removal plus descriptor stage runs at approximately 10–14 ms per keyframe in the reported sequences. Evaluation spans HeLiPR Town, INHA, and SemanticKITTI, with sensors including Ouster OS2-128, OS1-64, Velodyne VLP-16, Aeva Aeries II, Livox Avia, and Livox Mid-360.

Evaluation	Baseline or context	Uni-Mapper result
KITTI 00 dynamic removal	common voxel leaf size 0.1 m	SA 98.11, DA 89.99, AA 93.96, runtime 10 ms
HeLiPR Ouster1–Aeva2 loop matching	at 100% precision; STD gives 16 TP	23 TP, F1 0.660
HeLiPR Ouster1–Aeva2 map merging	LT-mapper (STD) RMSE 25.511	ATE RMSE 20.857
INHA WHEEL–DOG map merging	LT-mapper (SC) 7.663; LTA-OM 1.397	ATE RMSE 0.951

Selected alignment metrics further show improvement in AC and CD. For HeLiPR Ouster1–Aeva2, LT-STD reports AC $X$15 and CD $X$16, whereas Uni-Mapper reports AC $X$17 and CD $X$18. For INHA WHEEL–HAND1, LT-STD reports AC $X$19 and CD $X$20, LTA-OM reports AC $X$21 and CD $X$22, and Uni-Mapper reports AC $X$23 and CD $X$24. An ablation on HeLiPR Ouster1–Aeva2 reports: w/o All, ATE $X$25, AC $X$26, CD $X$27, MME $X$28; w/o DOR, ATE $X$29, AC $X$30, CD $X$31, MME $X$32; w/o REG, ATE $X$33, AC $X$34, CD $X$35, MME $X$36; full Uni-Mapper, ATE $X$37, AC $X$38, CD $X$39, MME $X$40. The paper concludes that two-stage registration delivers the largest gain, while dynamic object removal is crucial when initial loops are scarce or dynamic clutter dominates.

Across these literatures, Uni-Mapper does not denote a single method so much as a family of unifying constructions. In TDA, it names either the single-filter Mapper or a probabilistic, learned cover for Mapper, with the 1-dimensional theory clarifying structure, stability, and convergence to the Reeb graph. In robotics, it names either a standardized way to organize heterogeneous maps or a specific LiDAR map-merging framework that couples dynamic filtering, cross-modal place recognition, and anchor-based pose-graph optimization.