Euclidean Distance Matrix Completion

• Euclidean Distance Matrix Completion is the process of reconstructing a full, symmetric, and hollow distance matrix from partial data while enforcing low-rank and geometric constraints.
• Methods include low-rank SVD techniques, Gram matrix recovery, and semidefinite as well as manifold optimization to robustly capture the underlying Euclidean structure.
• Theoretical guarantees link coherence conditions and sample complexity to exact recovery, with extensions addressing noise, outliers, and structural constraints in real-world applications.

Searching arXiv for recent and foundational work on Euclidean Distance Matrix Completion. arXiv search: "Euclidean Distance Matrix Completion" Euclidean Distance Matrix Completion (EDMC) is the problem of recovering a full Euclidean distance matrix from a partially specified matrix of pairwise distances, or deciding whether such a completion exists in a prescribed embedding dimension. In its standard form, one seeks a symmetric hollow matrix $D$ with entries $D_{ij}=\|x_i-x_j\|^2$ for points $x_1,\dots,x_n\in\mathbb R^d$, while enforcing the characteristic geometric constraints that distinguish EDMs from generic low-rank matrices (Dokmanic et al., 2015, Fomin et al., 19 Mar 2026).

1. Geometric and algebraic structure

For $n$ points in $\mathbb R^d$, the EDM $D\in\mathbb R^{n\times n}$ is defined by

$D_{ij}=\|x_i-x_j\|^2.$

Such matrices are symmetric and satisfy $D_{ii}=0$. Two rank characterizations are used throughout the literature. First, $\operatorname{rank}(D)\le d+2$. Second, if

$J=I_n-\tfrac1n\mathbf1\mathbf1^T,$

then

$D_{ij}=\|x_i-x_j\|^2$0

is positive semidefinite and has rank at most $D_{ij}=\|x_i-x_j\|^2$1 (Moreira et al., 2017, An et al., 2022).

The centered matrix $D_{ij}=\|x_i-x_j\|^2$2 is the Gram matrix of a centered realization, and conversely

$D_{ij}=\|x_i-x_j\|^2$3

This equivalence underlies most completion methods: some work directly in distance space, while others recover a low-rank positive semidefinite Gram matrix and then map it back to distances (Dokmanic et al., 2015, Smith et al., 31 Jul 2025).

The feasible set of all EDMs forms the Euclidean distance cone $D_{ij}=\|x_i-x_j\|^2$4. Given a graph $D_{ij}=\|x_i-x_j\|^2$5 indexing observed entries, the projection $D_{ij}=\|x_i-x_j\|^2$6 is exactly the set of partial matrices that admit completion. A notable geometric fact is that $D_{ij}=\|x_i-x_j\|^2$7 is always a closed convex cone in $D_{ij}=\|x_i-x_j\|^2$8. Under a chordality assumption, the minimal face containing the feasible region can be described combinatorially through the maximal cliques of the observation graph (Drusvyatskiy et al., 2014).

2. Formulations of the completion problem

The most direct formulation imposes fidelity on the observed entries and EDM structure on the completion. If $D_{ij}=\|x_i-x_j\|^2$9 is the observation set and $x_1,\dots,x_n\in\mathbb R^d$0 the sampling operator, a basic low-rank model is

$x_1,\dots,x_n\in\mathbb R^d$1

with $x_1,\dots,x_n\in\mathbb R^d$2 for a $x_1,\dots,x_n\in\mathbb R^d$3-dimensional EDM (Moreira et al., 2017). In this representation, EDM completion resembles classical matrix completion, but the rank bound alone does not capture all Euclidean geometry.

A second family of formulations uses the Gram matrix $x_1,\dots,x_n\in\mathbb R^d$4, $x_1,\dots,x_n\in\mathbb R^d$5, and the linear measurement model

$x_1,\dots,x_n\in\mathbb R^d$6

This yields rank-minimization or nuclear-norm programs such as

$x_1,\dots,x_n\in\mathbb R^d$7

and its convex relaxation obtained by replacing $x_1,\dots,x_n\in\mathbb R^d$8 with $x_1,\dots,x_n\in\mathbb R^d$9. Because $n$0, the nuclear norm reduces to $n$1, giving a semidefinite program (Tasissa et al., 2018).

A third formulation operates directly on coordinates. The tutorial literature emphasizes least-squares criteria such as the $n$2-stress objective

$n$3

which is nonconvex in the point coordinates but enforces the Euclidean model by construction (Dokmanic et al., 2015). In application-specific variants, additional constraints appear. In rigid-body localization, for example, missing tag-anchor distances are completed within elementwise lower and upper bounds $n$4, and the cost is written as

$n$5

subject to symmetry, zero diagonal, positive semidefiniteness of $n$6, rank constraints, and box bounds (An et al., 2022).

These formulations differ in how they encode identifiability. Whenever the embedding is recoverable, it is typically unique only up to rigid motion. For chordal observation graphs, facial reduction theory shows that one round of clique-based reduction restores strict feasibility and that the singularity degree is at most $n$7 (Drusvyatskiy et al., 2014).

3. Algorithmic families

A large class of methods treats EDMC as low-rank completion with EDM-aware postprocessing. A representative example is Fixed-Rank Soft-Impute, which iteratively forms

$n$8

computes an SVD, chooses $n$9, soft-thresholds the singular values, and enforces rank at most $\mathbb R^d$0 (Moreira et al., 2017). Closely related is the SVD-Reconstruct stage of SVD-MDS: missing entries are filled by a scaled truncated SVD of an unbiased estimator, followed by classical multidimensional scaling to recover coordinates (Zhang et al., 2018).

Another line of work embeds EDM-specific projection directly into the iteration. In ad hoc microphone array calibration, the proposed algorithm alternates an OptSpace-style low-rank update with projection onto the EDM cone, including symmetry, zero-diagonal, nonnegativity, and an explicit projection onto the Euclidean distance space. The stated purpose is to confine the recovered matrix to the EDM cone at each iteration (Taghizadeh et al., 2014). The tutorial literature also describes alternating rank enforcement, coordinate-wise descent for $\mathbb R^d$1-stress, and semidefinite relaxation implemented through the Gram matrix (Dokmanic et al., 2015).

Semidefinite and facial-reduction methods exploit clique structure in the observation graph. One algorithm relates cliques to exposed faces of the PSD cone, sums clique-based exposing matrices, and solves a reduced least-squares problem on the resulting face. A second method formulates max-trace semidefinite recovery with a constrained misfit, then solves a Pareto-frontier problem by inexact Newton iterations coupled with Frank–Wolfe (Drusvyatskiy et al., 2014). These methods are motivated by the observation that clique information can sharply reduce ambient dimension before numerical optimization.

More recent work uses fixed-rank manifold optimization. In quotient-manifold formulations, one writes $\mathbb R^d$2 with $\mathbb R^d$3, optimizes over equivalence classes $\mathbb R^d$4 for $\mathbb R^d$5, and computes Riemannian gradients and Hessians on the resulting manifold (Marić et al., 2020). Related EDG formulations optimize directly on the manifold of rank-$\mathbb R^d$6 PSD Gram matrices using tangent-space projections and retractions by best rank-$\mathbb R^d$7 approximation (Smith et al., 2024, Smith et al., 31 Jul 2025). A distinct Burer–Monteiro route is the Asymmetric Projected Gradient Descent algorithm, which uses a pseudo-gradient in Gram space together with row-wise trimming to enforce incoherence (Li et al., 28 Apr 2025).

4. Recovery guarantees and sample complexity

Theoretical analysis has proceeded along several tracks. For random EDMs, coherence bounds connect Euclidean geometry to matrix-completion theory. One result shows that when the point coordinates are drawn i.i.d. from an atomless law with finite fourth moments, the resulting EDM satisfies the coherence assumptions needed by the Candès–Recht theorem with high probability, yielding sufficient conditions for exact recovery from a limited number of uniformly sampled entries (Kalogerias et al., 2013).

Exact recovery for Gram-matrix formulations has been established without relying on restricted isometry in the standard entrywise sense. In one convex theory, the distance measurements are recast as linear measurements of a centered Gram matrix in a non-orthogonal basis, and exact recovery is proved under coherence conditions when

$\mathbb R^d$8

uniformly random samples are available (Tasissa et al., 2018). A later Riemannian theory proves local linear convergence for rank-$\mathbb R^d$9 optimization under Bernoulli sampling when

$D\in\mathbb R^{n\times n}$0

with a one-step hard-thresholding initializer that is sufficient under

$D\in\mathbb R^{n\times n}$1

(Smith et al., 31 Jul 2025). A related 2024 analysis establishes linear convergence of a Riemannian gradient-like method with $D\in\mathbb R^{n\times n}$2, improved to $D\in\mathbb R^{n\times n}$3 using a refined resampled initialization (Smith et al., 2024).

The performance limits of simpler SVD-based estimators are also understood. For SVD-Reconstruct, expectation and high-probability bounds are available for $D\in\mathbb R^{n\times n}$4, together with propagation bounds for the MDS embedding error. The same analysis gives a minimax lower bound for zero-diagonal symmetric low-rank matrix completion, showing that SVD-Reconstruct is suboptimal in the minimax sense in the low-noise regime, while in the high-noise regime it achieves the optimal rate up to a constant factor (Zhang et al., 2018).

Some guarantees are explicitly algorithmic. The APGD analysis gives global linear convergence with exact recovery from

$D\in\mathbb R^{n\times n}$5

Bernoulli random observations without sample splitting (Li et al., 28 Apr 2025). For rigid-body localization via EDM completion, the completion subroutine SQREDM is described as an $D\in\mathbb R^{n\times n}$6-per-iteration majorization–minimization method, while the paper contrasts the overall pipeline with semidefinite-relaxation baselines costing $D\in\mathbb R^{n\times n}$7 in the worst case (An et al., 2022).

5. Noise, outliers, and structured variants

A substantial part of the literature addresses deviations from the ideal random-sampling model. In harsh-environment AGV localization, missing time-of-flight measurements are converted into missing EDM entries, and geometry-derived box bounds are used to constrain the unknown distances. Specifically, bounds for missing tag-anchor distances are derived from triangle inequalities, measured tag-anchor ranges $D\in\mathbb R^{n\times n}$8, and known inter-tag distances. The completed EDM is then embedded by classical MDS, refined by a weighted Gauss–Newton step, and converted to vehicle pose by Procrustes initialization followed by Gauss–Newton refinement (An et al., 2022). The reported EDM-completion error remains $D\in\mathbb R^{n\times n}$9 with the proposed bounds versus $D_{ij}=\|x_i-x_j\|^2.$0 for a naive shortest-path bound, and the final solver runs in $D_{ij}=\|x_i-x_j\|^2.$1 versus $D_{ij}=\|x_i-x_j\|^2.$2 for SDR, while staying $D_{ij}=\|x_i-x_j\|^2.$3 available in harsh scenes with many missing TOFs (An et al., 2022).

Outlier-robust completion has produced several distinct models. One approach writes the observed partial matrix as

$D_{ij}=\|x_i-x_j\|^2.$4

where $D_{ij}=\|x_i-x_j\|^2.$5 is a sparse outlier matrix, and jointly optimizes over coordinates, distances, and outliers by alternating updates of $D_{ij}=\|x_i-x_j\|^2.$6, a deep-neural-network parameterization of $D_{ij}=\|x_i-x_j\|^2.$7, and soft-thresholding of $D_{ij}=\|x_i-x_j\|^2.$8 (Kim, 17 Aug 2025). In the reported 3D-IoT experiments with $D_{ij}=\|x_i-x_j\|^2.$9, $D_{ii}=0$0, DNN depth $D_{ii}=0$1, and 1000 Monte-Carlo trials, the method cuts MSLE by approximately $D_{ii}=0$2 relative to the no-outlier baseline at outlier ratio $D_{ii}=0$3 (Kim, 17 Aug 2025). A different robust model augments Riemannian localization with an $D_{ii}=0$4-penalized outlier matrix and solves the resulting problem by alternating minimization (Nguyen et al., 2017).

Bayesian EDMC addresses sparsity and noise by placing priors directly on the latent point set. A hierarchical model with Gaussian point priors, a Normal–Wishart hyperprior, and a Gamma prior on the noise precision performs posterior inference by Metropolis–Hastings within Gibbs. Because the sampler operates in point space, the implied Gram matrix is always PSD and the induced distances automatically satisfy triangle inequalities, PSD-centered structure, and rank bounds (Chiluvuri et al., 30 Jan 2026). On synthetic data with $D_{ii}=0$5, $D_{ii}=0$6, observation fraction $D_{ii}=0$7, and SNR around $D_{ii}=0$8 dB, the method is reported to achieve up to $D_{ii}=0$9 lower relative error than OptSpace or alternating descent in sparse regimes (Chiluvuri et al., 30 Jan 2026).

Specialized structural constraints generate further EDMC variants. In the minimal-spanning-tree-only setting, the goal is to complete the matrix while preserving the given MST. The constructive guided-random-search algorithm described for this regime places points incrementally and checks MST consistency at each step; the paper reports that this constructive method clearly outperforms the compared standard EDMCP methods and is the only one that always preserves the MST by design (Rahman et al., 2016). In robotics, inverse kinematics has been reformulated as low-rank EDM completion by representing rigidly attached points through a partially observed EDM and then optimizing a rank-constrained Gram matrix on a quotient manifold (Marić et al., 2020).

6. Exact complexity, misconceptions, and current directions

The decision version of EDMC has recently received a parameterized-complexity treatment. For $\operatorname{rank}(D)\le d+2$0-EDMC, there is an FPT algorithm parameterized by $\operatorname{rank}(D)\le d+2$1 and the maximum number of unspecified entries per row or column, based on a compression procedure that reduces an instance to a principal submatrix of size at most $\operatorname{rank}(D)\le d+2$2, yielding running time

$\operatorname{rank}(D)\le d+2$3

A second FPT result is parameterized by $\operatorname{rank}(D)\le d+2$4 and the minimum number of fully specified principal submatrices whose entries cover all specified entries, and a further result gives polynomial-time solvability when both $\operatorname{rank}(D)\le d+2$5 and the minimum fill-in of the observation graph are fixed constants (Fomin et al., 19 Mar 2026). The same work also shows that, for every $\operatorname{rank}(D)\le d+2$6, $\operatorname{rank}(D)\le d+2$7-EDMC remains strongly NP-hard even if at most $\operatorname{rank}(D)\le d+2$8 entries of each row are unspecified (Fomin et al., 19 Mar 2026).

Several recurring misconceptions are corrected by the modern literature. One is that EDMC is merely generic low-rank matrix completion with a symmetric hollow matrix. The Gram-based formulations show otherwise: the measurement operator is not entrywise, the relevant basis is non-orthogonal, and standard restricted isometry can fail in this setting, motivating dual-basis analyses instead (Tasissa et al., 2018). Another misconception is that Euclidean constraints are automatically captured by low rank alone. EDM-specific conditions such as positive semidefiniteness of $\operatorname{rank}(D)\le d+2$9, triangle inequalities, centering, and cone geometry are explicitly used in semidefinite, projection, and Bayesian approaches (Drusvyatskiy et al., 2014, Chiluvuri et al., 30 Jan 2026).

Current directions reflect both theoretical and algorithmic gaps. The low-rank Soft-Impute literature notes sensitivity to the parameter $J=I_n-\tfrac1n\mathbf1\mathbf1^T,$0 and identifies automatic selection and more efficient randomized SVD or sketching as open directions (Moreira et al., 2017). The APGD analysis raises questions about why the pseudo-gradient requires substantially more samples for stabilization in practice and whether a refined incoherence notion or a better preconditioner could remove the need for explicit trimming (Li et al., 28 Apr 2025). Riemannian EDG work reports that optimizing over manifolds of higher-than-rank-$J=I_n-\tfrac1n\mathbf1\mathbf1^T,$1 matrices can yield superior numerical results, which suggests an interaction between overparameterization and nonconvex geometry that remains theoretically incomplete (Smith et al., 2024). Taken together, these strands indicate that EDMC sits at the intersection of distance geometry, semidefinite optimization, low-rank recovery, and exact graph-structural algorithms, with no single formulation dominating across all sampling regimes, noise models, and application constraints.