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Euclidean Distance Matrix Completion via Asymmetric Projected Gradient Descent

Yicheng Li, Xinghua Sun

TL;DR

The paper addresses recovering a low-rank EDM from partial distance data by introducing Asymmetric Projected Gradient Descent (APGD) with Burer-Monteiro factorization. It establishes global convergence guarantees under a Bernoulli sampling model without splitting data, relying on a pre-conditioned sampling operator that accounts for the non-diagonal EDM basis. The main contributions include a contraction-based convergence theorem, two novel uniform residual bounds, and an OS-MDS initialization that enables entering the contraction region, supported by extensive numerical experiments showing near-linear convergence in rich-sample regimes and phase-transition behavior under varying sample rates and perturbations. The results illuminate the effectiveness and limitations of non-convex EDMC methods, revealing the nuanced role of incoherence and implicit regularization in practical recovery tasks. Overall, the work advances theoretical understanding and practical performance of non-convex EDMC approaches with provable guarantees and insightful empirical validation.

Abstract

This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization, called the Asymmetric Projected Gradient Descent (APGD), for reconstructing the point set configuration from partial Euclidean distance measurements, known as the Euclidean Distance Matrix Completion (EDMC) problem. By paralleling the incoherence matrix completion framework, we show for the first time that global convergence guarantee with exact recovery of this routine can be established given $\mathcal{O}(μ^2 r^3 κ^2 n \log n)$ Bernoulli random observations without any sample splitting. Unlike leveraging the tangent space Restricted Isometry Property (RIP) and local curvature of the low-rank embedding manifold in some very recent works, our proof provides extra upper bounds that act as analogies of the random graph lemma under EDMC setting. The APGD works surprisingly well and numerical experiments demonstrate exact linear convergence behavior in rich-sample regions yet deteriorates rapidly when compared with the performance obtained by optimizing the s-stress function, i.e., the standard but unexplained non-convex approach for EDMC, if the sample size is limited. While virtually matching our theoretical prediction, this unusual phenomenon might indicate that: (i) the power of implicit regularization is weakened when specified in the APGD case; (ii) the stabilization of such new gradient direction requires substantially more samples than the information-theoretic limit would suggest.

Euclidean Distance Matrix Completion via Asymmetric Projected Gradient Descent

TL;DR

The paper addresses recovering a low-rank EDM from partial distance data by introducing Asymmetric Projected Gradient Descent (APGD) with Burer-Monteiro factorization. It establishes global convergence guarantees under a Bernoulli sampling model without splitting data, relying on a pre-conditioned sampling operator that accounts for the non-diagonal EDM basis. The main contributions include a contraction-based convergence theorem, two novel uniform residual bounds, and an OS-MDS initialization that enables entering the contraction region, supported by extensive numerical experiments showing near-linear convergence in rich-sample regimes and phase-transition behavior under varying sample rates and perturbations. The results illuminate the effectiveness and limitations of non-convex EDMC methods, revealing the nuanced role of incoherence and implicit regularization in practical recovery tasks. Overall, the work advances theoretical understanding and practical performance of non-convex EDMC approaches with provable guarantees and insightful empirical validation.

Abstract

This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization, called the Asymmetric Projected Gradient Descent (APGD), for reconstructing the point set configuration from partial Euclidean distance measurements, known as the Euclidean Distance Matrix Completion (EDMC) problem. By paralleling the incoherence matrix completion framework, we show for the first time that global convergence guarantee with exact recovery of this routine can be established given Bernoulli random observations without any sample splitting. Unlike leveraging the tangent space Restricted Isometry Property (RIP) and local curvature of the low-rank embedding manifold in some very recent works, our proof provides extra upper bounds that act as analogies of the random graph lemma under EDMC setting. The APGD works surprisingly well and numerical experiments demonstrate exact linear convergence behavior in rich-sample regions yet deteriorates rapidly when compared with the performance obtained by optimizing the s-stress function, i.e., the standard but unexplained non-convex approach for EDMC, if the sample size is limited. While virtually matching our theoretical prediction, this unusual phenomenon might indicate that: (i) the power of implicit regularization is weakened when specified in the APGD case; (ii) the stabilization of such new gradient direction requires substantially more samples than the information-theoretic limit would suggest.
Paper Structure (25 sections, 12 theorems, 85 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 25 sections, 12 theorems, 85 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Motivations
  3. Related Work
  4. Major Contributions and Organization
  5. Mathematical Setup and Main Result
  6. Main Result
  7. Proof of Theorem \ref{['thm_main_global_contraction']}
  8. Why APGD and Why not $\mathcal{R}_{\Omega}^*\mathcal{R}_{\Omega}$
  9. Numerical Experiments
  10. Implementation Details
  11. Phase Transition Under Gaussian Point Set
  12. Phase Transition Under Random Perturbation
  13. Iteration Trajectory
  14. Protein Test
  15. Conclusion and Discussion
  16. ...and 10 more sections

Key Result

Theorem 2.1

Under the above set up, let $p\gtrsim C_{\beta}\frac{\mu^2 r^3 \kappa^2\log n}{n}$. For any $k\geq 1$, i.e., Line 1 to Line 5 of Algorithm alg_APGD_Resample, we have holds with probability at least $1-cn^{1-\beta}-n^{2-\beta}$. Where $0<\eta^{\prime}\leq\frac{1}{336c_I\mu r\kappa^2}$, $c_c=\frac{80}{7}$, and $\Delta_{k}$ refers substituting $\mathbf{P}_k$ into eq_goedist_quotient. To reach $\vars

Figures (5)

  • Figure 1: The phase transition of regularization free OS-MDS and OS-MDS-GD when varying $p$ and $n$YCLiSunEDMC_Spec_init. We claim a success if the spectral error falls below $1$ in (a), or the EDM recover rate is smaller than $10^{-3}$ in (b). For each $(n,p)$, the result is obtained by 100 independent trials.
  • Figure 2: (a) The phase transition of APGD when varying Bernoulli sample rate $p$ and the number of points $n$. The result is obtained by 100 independent trials. To verify Theorem \ref{['thm_main_global_contraction']}, we claim a success if the EDM recover rate falls below $10^{-3}$. In (b), (c), and (d), we vary the constant $c_{IP}$ in \ref{['eq_incohrent_projection_C_I']} when performing incoherent projection, while all other parameters remain unchanged. (e) depicts the phase transition of IFHT-EDMC algorithm SmithCaiTasissaRieEDMC2 under the same problem setup as in (b). Subplots (f), (g), and (h) show selected phase transition curves when $n=500, 1000, 1500$ with $c_{IP}$ varies.
  • Figure 3: The phase transition of APGD when varying Bernoulli sample rate $p$ and the standard variance of perturbation white noise $\sigma_n$. A success is recorded if the EDM recover rate falls below $10^{-5}$. In (a), (b), and (c), we change the number of points from $n=500$ to $n=1500$, with all other parameters fixed. (d) plots the relationship between normalized quotient distance $\Vert\Delta\Vert_F^2/\sigma_r^{\star}$ and the intensity of noise $\sigma_n^2$.
  • Figure 4: Trajectories of all three quantities in \ref{['eq_record_track_ingrd']} while fixing $n=1500$. The result is obtained from 100 independent trials. (a), (d) shows the tendency of $r^k$ in selected succeed instances (96% when $p=0.1$, 47% when $p=0.05$). (b), (c) plots the corresponding $g_1^k$, $g_2^k$ when $p=0.1$. (e), (f) contains the records of $g_1^k$ and $g_2^k$ of selected failed instances (53% when $p=0.05$). These trajectories can be divided into two groups (17% in region 1, 36% in region 2).
  • Figure 5: The phase transition measured by EDM recovery rate of APGD when varying protein type and Bernoulli sample rate $p$, while fixing $r=3$. Detailed parameters for each protein are listed in Table \ref{['table_para_of_protein']}. The results are averaged over $5$ independent trials.

Theorems & Definitions (17)

  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Remark 2.2
  • Claim B.1
  • Lemma B.1
  • Lemma B.2
  • ...and 7 more