Huikang Liu

Huikang Liu, Man-Chung Yue, Anthony Man-Cho So

An estimation problem of fundamental interest is that of phase synchronization, in which the goal is to recover a collection of phases using noisy measurements of relative phases. It is known that in the Gaussian noise setting, the maximum likelihood estimator (MLE) has an expected squared $\ell_2$-estimation error that is on the same order as the Cramér-Rao lower bound. Moreover, even though the MLE is an optimal solution to a non-convex quadratic optimization problem, it can be found with high probability using semidefinite programming (SDP), provided that the noise power is not too large. In this paper, we study the estimation and convergence performance of a recently-proposed low-complexity alternative to the SDP-based approach, namely, the generalized power method (GPM). Our contribution is twofold. First, we bound the rate at which the estimation error decreases in each iteration of the GPM and use this bound to show that all iterates---not just the MLE---achieve an estimation error that is on the same order as the Cramér-Rao bound. Our result holds under the least restrictive assumption on the noise power and gives the best provable bound on the estimation error known to date. It also implies that one can terminate the GPM at any iteration and still obtain an estimator that has a theoretical guarantee on its estimation error. Second, we show that under the same assumption on the noise power as that for the SDP-based method, the GPM will converge to the MLE at a linear rate with high probability. This answers a question raised in [3] and shows that the GPM is competitive in terms of both theoretical guarantees and numerical efficiency with the SDP-based method. At the heart of our convergence rate analysis is a new error bound for the non-convex quadratic optimization formulation of the phase synchronization problem, which could be of independent interest.

Jiajin Li, Jianheng Tang, Lemin Kong et al.

In this work, we present the Bregman Alternating Projected Gradient (BAPG) method, a single-loop algorithm that offers an approximate solution to the Gromov-Wasserstein (GW) distance. We introduce a novel relaxation technique that balances accuracy and computational efficiency, albeit with some compromises in the feasibility of the coupling map. Our analysis is based on the observation that the GW problem satisfies the Luo-Tseng error bound condition, which relates to estimating the distance of a point to the critical point set of the GW problem based on the optimality residual. This observation allows us to provide an approximation bound for the distance between the fixed-point set of BAPG and the critical point set of GW. Moreover, under a mild technical assumption, we can show that BAPG converges to its fixed point set. The effectiveness of BAPG has been validated through comprehensive numerical experiments in graph alignment and partition tasks, where it outperforms existing methods in terms of both solution quality and wall-clock time.

Huikang Liu, Aras Selvi, Wolfram Wiesemann

We design a class of additive noise mechanisms that satisfy \((\varepsilon, δ)\)-differential privacy (DP) for scalar, real-valued query functions with known sensitivities, with a particular focus on moderate and low-privacy regimes. These mechanisms, which we call \textit{mixture mechanisms}, are constructed by mixing multiple Gaussian distributions that share the same variance but differ in their means and mixture weights. The resulting distributions can be interpreted as convex combinations of a zero-mean Gaussian (as used in the analytic Gaussian mechanism) and additional Gaussians whose means depend on the sensitivity of the query function. We derive tight conditions on the variances required for \((\varepsilon, δ)\)-DP and provide efficient algorithms to compute them. Compared to the analytic Gaussian mechanism, our mechanisms yield substantially lower expected noise amplitudes (\(l_1\)-loss) and variances (\(l_2\)-loss for zero-mean distributions). In the low-privacy regime that motivates our design, our mechanisms approach optimality, mitigating nearly all of the optimality gap of the analytic Gaussian mechanism.

Aras Selvi, Huikang Liu, Wolfram Wiesemann

In recent years, differential privacy has emerged as the de facto standard for sharing statistics of datasets while limiting the disclosure of private information about the involved individuals. This is achieved by randomly perturbing the statistics to be published, which in turn leads to a privacy-accuracy trade-off: larger perturbations provide stronger privacy guarantees, but they result in less accurate statistics that offer lower utility to the recipients. Of particular interest are therefore optimal mechanisms that provide the highest accuracy for a pre-selected level of privacy. To date, work in this area has focused on specifying families of perturbations a priori and subsequently proving their asymptotic and/or best-in-class optimality. In this paper, we develop a class of mechanisms that enjoy non-asymptotic and unconditional optimality guarantees. To this end, we formulate the mechanism design problem as an infinite-dimensional distributionally robust optimization problem. We show that the problem affords a strong dual, and we exploit this duality to develop converging hierarchies of finite-dimensional upper and lower bounding problems. Our upper (primal) bounds correspond to implementable perturbations whose suboptimality can be bounded by our lower (dual) bounds. Both bounding problems can be solved within seconds via cutting plane techniques that exploit the inherent problem structure. Our numerical experiments demonstrate that our perturbations can outperform the previously best results from the literature on artificial as well as standard benchmark problems.

Jiajin Li, Jianheng Tang, Lemin Kong et al.

In this paper, we study the design and analysis of a class of efficient algorithms for computing the Gromov-Wasserstein (GW) distance tailored to large-scale graph learning tasks. Armed with the Luo-Tseng error bound condition~\citep{luo1992error}, two proposed algorithms, called Bregman Alternating Projected Gradient (BAPG) and hybrid Bregman Proximal Gradient (hBPG) enjoy the convergence guarantees. Upon task-specific properties, our analysis further provides novel theoretical insights to guide how to select the best-fit method. As a result, we are able to provide comprehensive experiments to validate the effectiveness of our methods on a host of tasks, including graph alignment, graph partition, and shape matching. In terms of both wall-clock time and modeling performance, the proposed methods achieve state-of-the-art results.

Hongpei Li, Yicheng Huang, Huikang Liu et al.

We present a distributed framework of the Primal-Dual Hybrid Gradient (PDHG) algorithm for solving massive-scale linear programming (LP) problems. Although PDHG-based solvers demonstrate strong performance on single-node GPU architectures, their applicability to industrial-scale instances is often limited by single-GPU computational throughput. To overcome these challenges, we propose D-PDLP, the first Distributed PDLP framework, which extends PDHG to a multi-GPU setting via a practical two-dimensional grid partitioning of the constraint matrix. To improve load balance and computational efficiency, we introduce a block-wise random permutation strategy combined with nonzero-aware matrix partitioning. By distributing the intensive computation required in PDHG iterations, the proposed framework harnesses multi-GPU parallelism to achieve substantial speedups with relatively low communication overhead. Extensive experiments on standard LP benchmarks (including MIPLIB and Mittelmann instances) as well as huge-scale real-world datasets show that our distributed implementation, built upon cuPDLPx, achieves strong scalability and high performance while preserving full FP64 numerical accuracy.

Hongpei Li, Han Zhang, Huikang Liu et al.

Pipeline parallelism (PP) has become a standard technique for scaling large language model (LLM) training across multiple devices. However, despite recent progress in reducing memory consumption through activation offloading, existing approaches remain largely heuristic and coarse-grained, often overlooking the fine-grained trade-offs between memory, computation, and scheduling latency. In this work, we revisit the pipeline scheduling problem from a principled optimization perspective. We observe that prevailing strategies either rely on static rules or aggressively offload activations without fully leveraging the interaction between memory constraints and scheduling efficiency. To address this, we formulate scheduling as a constrained optimization problem that jointly accounts for memory capacity, activation reuse, and pipeline bubble minimization. Solving this model yields fine-grained schedules that reduce pipeline bubbles while adhering to strict memory budgets. Our approach complements existing offloading techniques: whereas prior approaches trade memory for time in a fixed pattern, we dynamically optimize the tradeoff with respect to model structure and hardware configuration. Experimental results demonstrate that our method consistently improves both throughput and memory utilization. In particular, we reduce idle pipeline time by up to 50% under the same per-device memory limit, and in some cases, enable the training of larger models within limited memory budgets.

Wanyu Zhang, Jiaqi Zhang, Dongdong Ge et al.

This paper addresses the problem of vision-based pedestrian localization, which estimates a pedestrian's location using images and camera parameters. In practice, however, calibrated camera parameters often deviate from the ground truth, leading to inaccuracies in localization. To address this issue, we propose an anchor-based method that leverages fixed-position anchors to reduce the impact of camera parameter errors. We provide a theoretical analysis that demonstrates the robustness of our approach. Experiments conducted on simulated, real-world, and public datasets show that our method significantly improves localization accuracy and remains resilient to noise in camera parameters, compared to methods without anchors.

Huikang Liu, Peng Wang, Longxiu Huang et al.

We study the problem of symmetric positive semi-definite low-rank matrix completion (MC) with deterministic entry-dependent sampling. In particular, we consider rectified linear unit (ReLU) sampling, where only positive entries are observed, as well as a generalization to threshold-based sampling. We first empirically demonstrate that the landscape of this MC problem is not globally benign: Gradient descent (GD) with random initialization will generally converge to stationary points that are not globally optimal. Nevertheless, we prove that when the matrix factor with a small rank satisfies mild assumptions, the nonconvex objective function is geodesically strongly convex on the quotient manifold in a neighborhood of a planted low-rank matrix. Moreover, we show that our assumptions are satisfied by a matrix factor with i.i.d. Gaussian entries. Finally, we develop a tailor-designed initialization for GD to solve our studied formulation, which empirically always achieves convergence to the global minima. We also conduct extensive experiments and compare MC methods, investigating convergence and completion performance with respect to initialization, noise level, dimension, and rank.

Peng Wang, Huikang Liu, Druv Pai et al.

The maximal coding rate reduction (MCR$^2$) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima, as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR$^2$ problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR$^2$ a natural choice of objective for learning diverse and discriminative representations via first-order optimization methods. To validate our theoretical findings, we conduct extensive experiments on both synthetic and real data sets.

Huikang Liu, Xiao Li, Anthony Man-Cho So

This work presents ReSync, a Riemannian subgradient-based algorithm for solving the robust rotation synchronization problem, which arises in various engineering applications. ReSync solves a least-unsquared minimization formulation over the rotation group, which is nonsmooth and nonconvex, and aims at recovering the underlying rotations directly. We provide strong theoretical guarantees for ReSync under the random corruption setting. Specifically, we first show that the initialization procedure of ReSync yields a proper initial point that lies in a local region around the ground-truth rotations. We next establish the weak sharpness property of the aforementioned formulation and then utilize this property to derive the local linear convergence of ReSync to the ground-truth rotations. By combining these guarantees, we conclude that ReSync converges linearly to the ground-truth rotations under appropriate conditions. Experiment results demonstrate the effectiveness of ReSync.

Yanguang Chen, Wenzhi Gao, Wanyu Zhang et al.

In this paper, we propose a Pre-trained Mixed Integer Optimization framework (PreMIO) that accelerates online mixed integer program (MIP) solving with offline datasets and machine learning models. Our method is based on a data-driven multi-variable cardinality branching procedure that splits the MIP feasible region using hyperplanes chosen by the concentration inequalities. Unlike most previous ML+MIP approaches that either require complicated implementation or suffer from a lack of theoretical justification, our method is simple, flexible, provable, and explainable. Numerical experiments on both classical OR benchmark datasets and real-life instances validate the efficiency of our proposed method.

Huikang Liu, Man-Chung Yue, Anthony Man-Cho So

The problem of synchronization over a group $\mathcal{G}$ aims to estimate a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations of a subset of all pairwise ratios of the form $G^*_i {G^*_j}^{-1}$. Such a problem has gained much attention recently and finds many applications across a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems in which the group is a closed subgroup of the orthogonal group. This class covers many group synchronization problems that arise in practice. Our contribution is fivefold. First, we propose a unified approach for solving this class of group synchronization problems, which consists of a suitable initialization step and an iterative refinement step based on the generalized power method, and show that it enjoys a strong theoretical guarantee on the estimation error under certain assumptions on the group, measurement graph, noise, and initialization. Second, we formulate two geometric conditions that are required by our approach and show that they hold for various practically relevant subgroups of the orthogonal group. The conditions are closely related to the error-bound geometry of the subgroup -- an important notion in optimization. Third, we verify the assumptions on the measurement graph and noise for standard random graph and random matrix models. Fourth, based on the classic notion of metric entropy, we develop and analyze a novel spectral-type estimator. Finally, we show via extensive numerical experiments that our proposed non-convex approach outperforms existing approaches in terms of computational speed, scalability, and/or estimation error.

Huikang Liu, Weijie Wu, Anthony Man-Cho So

A fundamental class of matrix optimization problems that arise in many areas of science and engineering is that of quadratic optimization with orthogonality constraints. Such problems can be solved using line-search methods on the Stiefel manifold, which are known to converge globally under mild conditions. To determine the convergence rate of these methods, we give an explicit estimate of the exponent in a Lojasiewicz inequality for the (non-convex) set of critical points of the aforementioned class of problems. By combining such an estimate with known arguments, we are able to establish the linear convergence of a large class of line-search methods. A key step in our proof is to establish a local error bound for the set of critical points, which may be of independent interest.