Bidirectional Local Distance Overview

• Bidirectional Local Distance is a concept that bridges local metric information with global structure through symmetric, two-sided mechanisms.
• It is applied in domains like zigzag persistence, Euclidean embedding, 3D pose estimation, local differential privacy, and motion planning to enhance data integration and repair.
• Each field adapts the idea by using mechanisms such as interval restriction, symmetric neighborhood graphs, scan reordering, and common-neighbor repair to approximate or stabilize global metrics.

Searching arXiv for the cited papers to ground the article and confirm identifiers. “Bidirectional local distance” does not appear in the cited arXiv literature as a single standardized formal object. Instead, the phrase designates a family of technically distinct constructions in which local distance information is related to a two-sided, symmetric, forward–backward, or local-versus-global comparison mechanism. In zigzag persistence, the relevant result is a one-sided theorem comparing interval-restricted and unrestricted bottleneck distances. In Euclidean embedding from sparse graphs, the central object is a symmetric local distance graph whose local metric information is integrated into a globally consistent embedding. In monocular 3D human pose estimation, locality is implemented implicitly by geometry-aware scan order inside a bidirectional global-local spatio-temporal state space model. In graph privacy, local distance vectors are propagated across undirected neighborhoods under Local Differential Privacy. In motion planning, local bidirectional proximity appears through common-neighbor repair sets and local subset optimization for reconnecting forward and reverse trees (Gasparovic et al., 2019, Arabadjis, 19 May 2026, Huang et al., 2024, Sheng et al., 7 Aug 2025, Zhang et al., 27 Aug 2025).

1. Terminological status and principal usages

Across these works, the phrase refers less to a single metric than to recurring structural motifs: restriction from global to local windows, symmetry of neighborhood relations, bidirectional scanning, two-sided edge reporting, and local repair of forward–reverse search failures. Several of the papers explicitly state that they do not define a formal notion called “Bidirectional Local Distance,” and their relevance is therefore conceptual rather than terminological (Arabadjis, 19 May 2026, Huang et al., 2024, Sheng et al., 7 Aug 2025, Zhang et al., 27 Aug 2025).

Domain	Closest formulation	Directionality pattern
Zigzag persistence	local-versus-global bottleneck distance under interval restriction	one-sided inequality
Euclidean embedding	symmetric local distance on undirected edges	reciprocal neighborhood consistency
3D pose estimation	bidirectional global-local spatio-temporal SSM block	forward/backward scans
Local DP graph querying	local distance-vector aggregation	symmetric undirected propagation
Motion planning	local bidirectional connection repair and local subset optimization	forward–reverse tree grafting

A central misconception is to treat “bidirectional” as implying a theorem in both directions. The persistence result proves only that a restricted local bottleneck distance cannot exceed the unrestricted global bottleneck distance. Conversely, some of the other works are genuinely bidirectional in mechanism—forward and backward scans, two-sided neighborhood exchange, or simultaneous use of both endpoints of a failed edge—but do not define a distance function with formal bidirectional axioms (Gasparovic et al., 2019, Huang et al., 2024, Sheng et al., 7 Aug 2025, Zhang et al., 27 Aug 2025).

2. Zigzag persistence: local restrictions as lower bounds on global distance

In zigzag persistence, the core setting is a zigzag module

$$V_1 \leftrightarrow V_2 \leftrightarrow \cdots \leftrightarrow V_n$$

obtained, for example, by applying fixed-degree homology to a zigzag of spaces. A key structural fact is interval decomposition:

$$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$

with persistence diagram $Dg\,\mathbb X$ determined by the interval endpoints. Restricting to an interval $I=[r_1,r_2]$ clips each interval bar:

$$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$

The induced projection $\Pi$ sends a diagram point $(b,d)$ to its clipped version in $I$, with bars entirely outside the interval sent to diagonal points; these added diagonal points do not affect bottleneck-distance arguments (Gasparovic et al., 2019).

The main theorem is

$$d_B(Dg\,\mathbb X^I, Dg\,\mathbb Y^I)\le d_B(Dg\,\mathbb X, Dg\,\mathbb Y).$$

This is the paper’s precise local-versus-global statement. It establishes that interval restriction cannot increase bottleneck distance. The proof is diagrammatic: start with a global partial matching $\mu$, project matched pairs by $$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$0, obtain a valid restricted matching $$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$1, and show

$$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$2

The underlying intuition is that restriction clips bars inward toward the interval or sends them to the diagonal, and these operations cannot enlarge the $$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$3 matching cost (Gasparovic et al., 2019).

The result is one-sided rather than bidirectional. It proves

$$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$4

or equivalently that local distances are lower bounds on the global one. It does not prove a converse of the form

$$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$5

nor any general equivalence asserting that distances over all windows determine the global distance exactly. This is the most precise example in the cited literature of how “bidirectional local distance” can be misleading: the theorem relates local and global quantities in only one mathematical direction (Gasparovic et al., 2019).

The same monotonic principle extends to metric-graph persistence and multiparameter persistence. For geodesic distance functions $$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$6 on a metric graph, restriction to $$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$7 yields local neighborhood signatures, and the paper derives

$$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$8

For the local persistence distortion distance, if $$\mathbb X \cong \bigoplus_{j\in J} \mathbb I[b_j,d_j],$$9, then

$Dg\,\mathbb X$0

For $Dg\,\mathbb X$1-parameter modules, the matching-distance analogue is

$Dg\,\mathbb X$2

These extensions preserve the same structural message: local restriction yields stable lower-scale probes of global discrepancy, but not a converse control theorem (Gasparovic et al., 2019).

3. Euclidean embedding from symmetric local distances

In graph-based Euclidean embedding, the input is a neighborhood graph weighted by pairwise distances, without any prior vector representation of the data. The graph is a distance graph: an undirected graph whose edges are weighted by mutual distances between connected vertices. The target is a globally consistent Euclidean embedding $Dg\,\mathbb X$3 of an unknown manifold $Dg\,\mathbb X$4 into $Dg\,\mathbb X$5, where $Dg\,\mathbb X$6 is the intrinsic dimension. The paper is explicit that the approach fundamentally assumes “a symmetric relation between datapoints that are considered neighboring according to this relation,” and in experiments an $Dg\,\mathbb X$7-nearest-neighbor graph is symmetrized by

$Dg\,\mathbb X$8

If “bidirectional local distance” is interpreted as reciprocal local consistency, this symmetric graph assumption is the closest exact formulation (Arabadjis, 19 May 2026).

The method reconstructs local Euclidean frames from graph distances alone. Around each vertex $Dg\,\mathbb X$9, local geodesic distances between points in $I=[r_1,r_2]$0 are estimated by shortest paths in $I=[r_1,r_2]$1. If $I=[r_1,r_2]$2 is the matrix of squared geodesic distances, the local Gram matrix is

$I=[r_1,r_2]$3

and the local Euclidean embedding is recovered by

$I=[r_1,r_2]$4

These local frames are then converted into barycentric coordinates and a metric-trivializing infinitesimal transport $I=[r_1,r_2]$5, with the approximation

$I=[r_1,r_2]$6

The embedding differential is represented by

$I=[r_1,r_2]$7

and local mismatch is measured by

$I=[r_1,r_2]$8

where $I=[r_1,r_2]$9 is orthogonal, reflecting the fact that local metric structure is only determined up to rotation or reflection (Arabadjis, 19 May 2026).

The global objective is variational rather than edgewise:

$$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$0

The stationary pair satisfies

$$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$1

with $$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$2 obtained by a Procrustes/SVD update from the inner-product matrix between $$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$3 and $$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$4. The discrete implementation alternates sparse Poisson solves and local SVD updates, starting from $$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$5, with memory $$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$6 and per-iteration time

$$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$7

The main theorem states convergence of this iterative process to the equivalence class of global minimizers, unique up to constant rigid-body orthogonal transformations and constant translations of the embedding (Arabadjis, 19 May 2026).

This framework does not define a scalar “bidirectional local distance.” Its local distances are symmetric graph edge weights and locally computed shortest-path distances, while its global consistency arises from integrability of the differential field. The paper explicitly emphasizes that exact local metric conservation conflicts with global integrability on curved manifolds; the method therefore deforms local transports as little as necessary to achieve a globally coherent Euclidean embedding. That tension between local metric fidelity and global consistency is one of the clearest cross-domain meanings of the phrase (Arabadjis, 19 May 2026).

4. Bidirectional local scanning in monocular 3D human pose estimation

In monocular 3D human pose estimation, the closest analogue to “bidirectional local distance” is not an explicit metric but a scan-order mechanism inside a state space model. The paper proposes a “bidirectional global-local spatio-temporal SSM block” and states that “global refers to spatial modeling that captures the full-body pose” while “local pertains to spatial modeling focused on the limbs and their detailed movements.” Locality is enhanced by a “reordering strategy” that provides a “more logical geometric scanning order,” and this order is integrated with a global spatial scan to form a combined global-local spatial scan (Huang et al., 2024).

The architecture operates on 2D pose sequences

$$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$8

embedded as

$$\mathbb X[r_1,r_2] \cong \bigoplus_{j\in J} \mathbb I([b_j,d_j]\cap [r_1,r_2]).$$9

with spatial and temporal positional embeddings and final output

$\Pi$0

Its block update is

$\Pi$1

Bidirectionality enters through four reorganized scans before the SSM: forward spatial, forward temporal, backward spatial, and backward temporal. The local component is added because “relying only on global scanning consistently led to inaccurate limb prediction,” whereas the local scan is intended “to capture local human skeleton details” (Huang et al., 2024).

The paper does not define an explicit local distance variable, neighborhood radius, graph geodesic metric, or Euclidean threshold for locality. Its locality is implemented implicitly through token adjacency after reordering. The global scan is described as scanning joints “from 0 to 16,” while the local scan follows a geometry-aware ordering that makes anatomically related joints closer in the one-dimensional sequence processed by the SSM. A plausible implication is that the effective local relation is governed by sequence-order adjacency rather than an explicit metric, but the paper presents this mechanism conceptually and visually rather than through a formal distance function (Huang et al., 2024).

The empirical evidence is an ablation over scan strategies. Unidirectional spatio-temporal variants report 43.1, 43.2, 43.8, and 43.0; bidirectional spatio-temporal scanning reports 42.4; and bidirectional global-local spatio-temporal scanning reports 41.8. The supplementary discussion further states that the model “shines remarkably in assessing limb movements, particularly regarding the shoulders and elbows.” Here, the term “bidirectional local distance” is therefore best understood as a shorthand for bidirectional local scan structure rather than a scalar distance object (Huang et al., 2024).

5. Local distance propagation under Local Differential Privacy

In graph privacy, the relevant setting is a distributed graph in which each vertex privately holds its own neighbor list $\Pi$2, and the target query returns all-pairs graph distances. The graph is explicitly undirected for the main formulation, so the intended distance is symmetric shortest-path distance in an unweighted graph. “Local” has two meanings: the local privacy model, where no trusted curator holds the whole graph, and local distance information, where each vertex initially knows only its one-hop neighborhood (Sheng et al., 7 Aug 2025).

The first approach, Graph Aggregation, reconstructs a synthetic graph from randomized responses. Each undirected edge is effectively reported by both endpoints and then merged by

$\Pi$3

An improved version randomly mixes $\Pi$4 and $\Pi$5 using

$\Pi$6

This is a genuinely two-sided mechanism, but the paper states that synthetic-graph methods suffer from low utility for distance queries because shortest-path distances are highly sensitive to graph structural corruption (Sheng et al., 7 Aug 2025).

The main contribution is Neighbor Aggregation, a local distance-vector method. Each vertex $\Pi$7 initializes a local vector $\Pi$8 with distance $\Pi$9 to itself, distance $(b,d)$0 to neighbors, and threshold $(b,d)$1 to all others. Privatization is performed either by Laplace noise or randomized response over $(b,d)$2, with

$(b,d)$3

for the randomized-response variant. Thereafter, distance information is propagated by repeated neighbor aggregation:

$(b,d)$4

This is not a classical bidirectional shortest-path search. Rather, it is a synchronous, all-destination distance-vector relaxation in which information flows across undirected edges in both directions over successive iterations (Sheng et al., 7 Aug 2025).

The paper proves that Algorithm 3 satisfies $(b,d)$5-LDP for each edge and argues that, without noise, the update process reconstructs shortest paths exactly in a BFS-like manner. Its probabilistic analysis shows that repeated minima make Laplace noise strongly downward biased, whereas randomized response performs much better. The experiments therefore adopt randomized response, and the paper reports that, except for EIES, the error ordering is

$(b,d)$6

In this literature, “bidirectional local distance” is most accurately read as symmetric local distance propagation rather than as source-side and target-side searches meeting in the middle (Sheng et al., 7 Aug 2025).

6. Bidirectional motion planning and local subset optimization

In motion planning, the nearest counterpart is not a named distance but a mechanism for local repair of failed forward–reverse connections. The planner G3T* operates in asymmetric bidirectional sampling-based motion planning with a forward tree and a lazy reverse tree. When a connecting edge $(b,d)$7 fails full collision detection, standard lazy-reverse search may restart the reverse tree, wasting useful local structure near the failed connection. G3T* instead performs grafting by expanding the source and target endpoints of the invalid edge and attempting a local bypass through a common nearby state (Zhang et al., 27 Aug 2025).

Local proximity is defined by radius-based neighborhoods

$(b,d)$8

where

$(b,d)$9

The central bidirectional local object is the common-neighbor intersection

$I$0

Each $I$1 induces a two-edge bypass

$I$2

and the candidate set is

$I$3

Candidates are ranked by the heuristic key

$I$4

and the planner greedily selects the best edge-pair that passes lazy and then full checking. In this setting, local bidirectionality is the simultaneous use of both endpoints of a failed inter-tree edge and their shared local neighborhood (Zhang et al., 27 Aug 2025).

The same paper introduces local subset optimization through guided incremental local densification. Once a solution path exists, the informed set is

$I$5

and a beacon on the current path is chosen by

$I$6

Here “minimum Lebesgue measure” means minimizing the sum of the hypervolumes of the induced front and back local hyperspheroids. The resulting GuILD subsets and greedy GuILD subsets are used to densify sampling in locally promising regions, while the sampling distribution is dynamically adjusted by current and historical cost improvements,

$I$7

The planner claims probabilistic completeness and asymptotic optimality while using these local mechanisms to accelerate convergence and reduce solution cost (Zhang et al., 27 Aug 2025).

The paper therefore supplies a practically strong, but terminologically indirect, interpretation of “bidirectional local distance.” The closest formal ingredients are Euclidean neighborhood distance, common-neighbor overlap, and heuristic scoring of local bypass edge-pairs; there is no single scalar distance that alone measures forward–reverse tree separation (Zhang et al., 27 Aug 2025).

7. Cross-domain interpretation and recurrent misconceptions

A cross-domain view suggests that “bidirectional local distance” is best treated as a comparative research theme rather than a settled mathematical term. The most rigorous local/global statement among the cited works is the zigzag-persistence inequality

$I$8

which is explicitly one-sided. By contrast, the Euclidean-embedding work is bidirectional chiefly through symmetric undirected local distance graphs; the pose-estimation work is bidirectional through forward/backward scan directions and local token reordering; the Local Differential Privacy work is bidirectional through symmetric undirected exchange and two-sided edge reporting; and the motion-planning work is bidirectional through simultaneous use of both endpoints of a failed inter-tree edge (Gasparovic et al., 2019, Arabadjis, 19 May 2026, Huang et al., 2024, Sheng et al., 7 Aug 2025, Zhang et al., 27 Aug 2025).

Several misconceptions recur. First, “local distance” need not denote an explicit scalar metric: in PoseMamba it is an implicit consequence of scan order, whereas in motion planning it is distributed across Euclidean thresholds, common-neighbor sets, and heuristic scores (Huang et al., 2024, Zhang et al., 27 Aug 2025). Second, “bidirectional” does not always mean directed forward/backward path search: in the Local Differential Privacy setting the graph is undirected and the central mechanism is symmetric propagation of local distance vectors, not classical bidirectional BFS (Sheng et al., 7 Aug 2025). Third, local information does not generally determine global structure without additional integrability, matching, or connectivity conditions: the Euclidean-embedding paper enforces integrability through a variational problem on differentials, and the persistence paper explicitly does not prove any converse from local windows to global distance (Arabadjis, 19 May 2026, Gasparovic et al., 2019).

Taken together, these works support a technically precise but plural understanding of the term. In persistence, bidirectional local distance is best replaced by “local-versus-global distance monotonicity.” In representation-free embedding, it denotes symmetric local metric information integrated into a global map. In SSM-based pose estimation, it refers to bidirectional local scanning over geometry-aware orderings. In local privacy, it refers to symmetric local distance-vector exchange. In motion planning, it refers to local bidirectional connection repair and local subset optimization. The common thread is that local relational information is used to stabilize, approximate, or repair a larger global structure, but the mathematical form of that relation is domain-specific rather than universal (Gasparovic et al., 2019, Arabadjis, 19 May 2026, Huang et al., 2024, Sheng et al., 7 Aug 2025, Zhang et al., 27 Aug 2025).