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Ziye Ma

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6 papers

2 author rows

AAAI Conference 2025 Conference Paper

Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integrity

Tianqi Shen
Shaohua Liu
Jiaqi Feng
Ziye Ma
Ning An

Gaussian Splatting (GS) has emerged as a crucial technique for representing discrete volumetric radiance fields. It leverages unique parametrization to mitigate computational demands in scene optimization. This work introduces Topology-Aware 3D Gaussian Splatting (Topology-GS), which addresses two key limitations in current approaches: compromised pixel-level structural integrity due to incomplete initial geometric coverage, and inadequate feature-level integrity from insufficient topological constraints during optimization. To overcome these limitations, Topology-GS incorporates a novel interpolation strategy, Local Persistent Voronoi Interpolation (LPVI), and a topology-focused regularization term based on persistent barcodes, named PersLoss. LPVI utilizes persistent homology to guide adaptive interpolation, enhancing point coverage in low-curvature areas while preserving topological structure. PersLoss aligns the visual perceptual similarity of rendered images with ground truth by constraining distances between their topological features. Comprehensive experiments on three novel-view synthesis benchmarks demonstrate that Topology-GS outperforms existing methods in terms of PSNR, SSIM, and LPIPS metrics, while maintaining efficient memory usage. This study pioneers the integration of topology with 3D-GS, laying the groundwork for future research in this area.

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JMLR Journal 2025 Journal Article

Towards Optimal Branching of Linear and Semidefinite Relaxations for Neural Network Robustness Certification

Brendon G. Anderson
Ziye Ma
Jingqi Li
Somayeh Sojoudi

In this paper, we study certifying the robustness of ReLU neural networks against adversarial input perturbations. To diminish the relaxation error suffered by the popular linear programming (LP) and semidefinite programming (SDP) certification methods, we take a branch-and-bound approach to propose partitioning the input uncertainty set and solving the relaxations on each part separately. We show that this approach reduces relaxation error, and that the error is eliminated entirely upon performing an LP relaxation with a partition intelligently designed to exploit the nature of the ReLU activations. To scale this approach to large networks, we consider using a coarser partition whereby the number of parts in the partition is reduced. We prove that computing such a coarse partition that directly minimizes the LP relaxation error is NP-hard. By instead minimizing the worst-case LP relaxation error, we develop a closed-form branching scheme in the single-hidden layer case. We extend the analysis to the SDP, where the feasible set geometry is exploited to design a branching scheme that minimizes the worst-case SDP relaxation error. Experiments on MNIST, CIFAR-10, and Wisconsin breast cancer diagnosis classifiers demonstrate significant increases in the percentages of test samples certified. By independently increasing the input size and the number of layers, we empirically illustrate under which regimes the branched LP and branched SDP are best applied. Finally, we extend our LP branching method into a multi-layer branching heuristic, which attains comparable performance to prior state-of-the-art heuristics on large-scale, deep neural network certification benchmarks. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

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NeurIPS Conference 2023 Conference Paper

Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing

Ziye Ma
Javad Lavaei
Somayeh Sojoudi

Gradient descent (GD) is crucial for generalization in machine learning models, as it induces implicit regularization, promoting compact representations. In this work, we examine the role of GD in inducing implicit regularization for tensor optimization, particularly within the context of the lifted matrix sensing framework. This framework has been recently proposed to address the non-convex matrix sensing problem by transforming spurious solutions into strict saddles when optimizing over symmetric, rank-1 tensors. We show that, with sufficiently small initialization scale, GD applied to this lifted problem results in approximate rank-1 tensors and critical points with escape directions. Our findings underscore the significance of the tensor parametrization of matrix sensing, in combination with first-order methods, in achieving global optimality in such problems.

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ICML Conference 2023 Conference Paper

Over-parametrization via Lifting for Low-rank Matrix Sensing: Conversion of Spurious Solutions to Strict Saddle Points

Ziye Ma
Igor Molybog
Javad Lavaei
Somayeh Sojoudi

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

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AAAI Conference 2023 Conference Paper

Semidefinite Programming versus Burer-Monteiro Factorization for Matrix Sensing

Baturalp Yalçın
Ziye Ma
Javad Lavaei
Somayeh Sojoudi

Many fundamental low-rank optimization problems, such as matrix completion, phase retrieval, and robust PCA, can be formulated as the matrix sensing problem. Two main approaches for solving matrix sensing are based on semidefinite programming (SDP) and Burer-Monteiro (B-M) factorization. The former suffers from high computational and space complexities, whereas the latter may return a spurious solution due to the non-convexity of the problem. The existing theoretical guarantees for the success of these methods have led to similar conservative conditions, which may wrongly imply that these methods have comparable performances. In this paper, we shed light on some major differences between these two methods. First, we present a class of structured matrix completion problems for which the B-M methods fail with an overwhelming probability, while the SDP method works correctly. Second, we identify a class of highly sparse matrix completion problems for which the B-M method works and the SDP method fails. Third, we prove that although the B-M method exhibits the same performance independent of the rank of the unknown solution, the success of the SDP method is correlated to the rank of the solution and improves as the rank increases. Unlike the existing literature that has mainly focused on those instances of matrix sensing for which both SDP and B-M work, this paper offers the first result on the unique merit of each method over the alternative approach.

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AAAI Conference 2022 Conference Paper

Sharp Restricted Isometry Property Bounds for Low-Rank Matrix Recovery Problems with Corrupted Measurements

Ziye Ma
Yingjie Bi
Javad Lavaei
Somayeh Sojoudi

In this paper, we study a general low-rank matrix recovery problem with linear measurements corrupted by some noise. The objective is to understand under what conditions on the restricted isometry property (RIP) of the problem local search methods can find the ground truth with a small error. By analyzing the landscape of the non-convex problem, we first propose a global guarantee on the maximum distance between an arbitrary local minimizer and the ground truth under the assumption that the RIP constant is smaller than 1/2. We show that this distance shrinks to zero as the intensity of the noise reduces. Our new guarantee is sharp in terms of the RIP constant and is much stronger than the existing results. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. Next, we prove the strict saddle property, which guarantees the global convergence of the perturbed gradient descent method in polynomial time. The developed results demonstrate how the noise intensity and the RIP constant of the problem affect the landscape of the problem.

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