Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees

• URL: http://arxiv.org/abs/2410.06376v1
• Date: Tue, 8 Oct 2024 21:19:22 GMT
• Title: Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees
• Authors: Chandler Smith, HanQin Cai, Abiy Tasissa,
• Abstract summary: Two algorithms are proposed to solve the Euclidean Distance Geometry problem. First algorithm converges linearly to the true solution. Second algorithm demonstrates strong numerical performance on both synthetic and real data.
• Score: 6.422262171968397
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms grounded in the Riemannian optimization framework to address the EDG problem. Our approach formulates the problem as a low-rank matrix completion task over the Gram matrix, using partial measurements represented as expansion coefficients of the Gram matrix in a non-orthogonal basis. For the first algorithm, under a uniform sampling with replacement model for the observed distance entries, we demonstrate that, with high probability, a Riemannian gradient-like algorithm on the manifold of rank-$r$ matrices converges linearly to the true solution, given initialization via a one-step hard thresholding. This holds provided the number of samples, $m$, satisfies $m \geq \mathcal{O}(n^{7/4}r^2 \log(n))$. With a more refined initialization, achieved through resampled Riemannian gradient-like descent, we further improve this bound to $m \geq \mathcal{O}(nr^2 \log(n))$. Our analysis for the first algorithm leverages a non-self-adjoint operator and depends on deriving eigenvalue bounds for an inner product matrix of restricted basis matrices, leveraging sparsity properties for tighter guarantees than previously established. The second algorithm introduces a self-adjoint surrogate for the sampling operator. This algorithm demonstrates strong numerical performance on both synthetic and real data. Furthermore, we show that optimizing over manifolds of higher-than-rank-$r$ matrices yields superior numerical results, consistent with recent literature on overparameterization in the EDG problem.

Related papers

• A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval [56.67706781191521]
  In this work, we present a robust phase retrieval problem where the task is to recover an unknown signal. Our proposed oracle avoids the need for computationally spectral descent, using a simple gradient step and outliers.
  arXiv  Detail & Related papers  (2024-09-07T06:37:23Z)
• Low-complexity subspace-descent over symmetric positive definite manifold [9.346050098365648]
  We develop low-complexity algorithms for the minimization of functions over the symmetric positive definite (SPD) manifold. The proposed approach utilizes carefully chosen subspaces that allow the update to be written as a product of the Cholesky factor of the iterate and a sparse matrix.
  arXiv  Detail & Related papers  (2023-05-03T11:11:46Z)
• Rank-1 Matrix Completion with Gradient Descent and Small Random Initialization [15.127728811011245]
  We show that implicit regularization of GD plays a critical role in analysis. We observe that implicit regularization GD plays a critical role in affordable analysis.
  arXiv  Detail & Related papers  (2022-12-19T12:05:37Z)
• Semi-Supervised Subspace Clustering via Tensor Low-Rank Representation [64.49871502193477]
  We propose a novel semi-supervised subspace clustering method, which is able to simultaneously augment the initial supervisory information and construct a discriminative affinity matrix. Comprehensive experimental results on six commonly-used benchmark datasets demonstrate the superiority of our method over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-05-21T01:47:17Z)
• Nonlinear matrix recovery using optimization on the Grassmann manifold [18.655422834567577]
  We investigate the problem of recovering a partially observed high-rank clustering matrix whose columns obey a nonlinear structure such as a union of subspaces. We show that the alternating limit converges to a unique point using the Kurdyka-Lojasi property.
  arXiv  Detail & Related papers  (2021-09-13T16:13:13Z)
• Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization [59.87663954467815]
  Algorithm unfolding creates an interpretable and parsimonious neural network architecture by implementing each iteration of a model-based algorithm as a neural layer. In this paper, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we unroll a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph. Experimental results show that our unrolled network outperformed pure model-based graph classifiers, and achieved comparable performance to pure data-driven networks but using far fewer parameters.
  arXiv  Detail & Related papers  (2021-09-10T07:01:15Z)
• Near-Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time [46.31710224483631]
  We show how to bypass the main open question of Nelson and Nguyen (FOCS, 2013) regarding the logarithmic factors in the sketching dimension for existing constant factor approximation oblivious subspace embeddings. A key technique we use is an explicit mapping of Indyk based on uncertainty principles and extractors. For the fundamental problems of rank computation and finding a linearly independent subset of columns, our algorithms improve Cheung, Kwok, and Lau (JACM, 2013) and are optimal to within a constant factor and a $loglog(n)$-factor, respectively.
  arXiv  Detail & Related papers  (2021-07-16T19:34:10Z)
• On the Efficient Implementation of the Matrix Exponentiated Gradient Algorithm for Low-Rank Matrix Optimization [26.858608065417663]
  Convex optimization over the spectrahedron has important applications in machine learning, signal processing and statistics. We propose efficient implementations of MEG, which are tailored for optimization with low-rank matrices, and only use a single low-rank SVD on each iteration. We also provide efficiently-computable certificates for the correct convergence of our methods.
  arXiv  Detail & Related papers  (2020-12-18T19:14:51Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)
• Optimal Iterative Sketching with the Subsampled Randomized Hadamard Transform [64.90148466525754]
  We study the performance of iterative sketching for least-squares problems. We show that the convergence rate for Haar and randomized Hadamard matrices are identical, andally improve upon random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.
  arXiv  Detail & Related papers  (2020-02-03T16:17:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.