一种基于半定规划的社区发现算法的性能

Bundle Method Sketching for Low Rank Semidefinite Programming

In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimization problems and from the analysis of bundle methods. Under strong duality and strict complementarity of SDP, our algorithm produces primal and the dual sequences converging in feasibility at a rate of $\tilde{O}(1/ε)$ and in optimality at a rate of $\tilde{O}(1/ε^2)$. Moreover, our algorithm outputs a low rank representation of its approximate s…

结构块模型的半定规划

A Semidefinite Program for Structured Blockmodels

半定规划最近被开发的问题的社区检测，这可能被视为一个特殊情况下的随机块模型。在这里，我们开发了一个半定的程序，可以针对块模型的其他实例，如非连续网络和重叠社区。我们建立标签恢复稀疏设置，与社区检测的最近结果类似的条件。在数据不是由块模型生成的情况下，我们给出了一个oracle不等式，该不等式限制了相对于最佳块模型近似的超额风险。模拟社区检测，重叠的社区，和潜在的空间模型。

用于恢复d维（$d\geq 2$）离网频率的原子模最小化的精确半定规划公式

Precise Semidefinite Programming Formulation of Atomic Norm Minimization for Recovering d-Dimensional ($d\geq 2$) Off-the-Grid Frequencies

离网压缩感知（CS）的最新研究表明，在一定条件下，即使字典是连续的，也可以成功地从一些时域样本中恢复频谱稀疏的信号。特别地，在\cite{tang2012csotg}中提出了原子范数最小化算法，以恢复$1$维谱稀疏信号.然而，尽管现有的研究工作\cite{chi2013compressive}，它仍然是一个悬而未决的问题，如何制定一个等效的半正定程序的原子范数最小化恢复信号与$d$维（$d\geq 2$）离网频率。在本文中，我们解决了这个问题，提出了等价的半定规划公式的原子范数最小化恢复信号与$d$维（$d\geq 2$）离网频率。

Training Quantized Neural Networks to Global Optimality via Semidefinite Programming

Neural networks (NNs) have been extremely successful across many tasks in machine learning. Quantization of NN weights has become an important topic due to its impact on their energy efficiency, inference time and deployment on hardware. Although post-training quantization is well-studied, training optimal quantized NNs involves combinatorial non-convex optimization problems which appear intractable. In this work, we introduce a convex optimization strategy to train quantized NNs with polynomial activations. Our method leverages hidden convexity in two-layer neural networks from the recent lit…

混合方法：带有对角约束的结构半正定规划的低秩坐标下降法

The Mixing method: low-rank coordinate descent for semidefinite programming with diagonal constraints

本文提出了一种低秩坐标下降方法，用于求解带对角约束的结构半正定规划问题。该方法在优化性能上显著优于现有方法，并证明其在随机初始化下以局部线性速率收敛到全局最优解。

半定规划用于亚高斯混合聚类

Clustering subgaussian mixtures by semidefinite programming

我们介绍了一个无模型的松弛和轮算法的k-均值聚类的基础上半定松弛由于彭和魏。该算法将SDP输出解释为原始数据的去噪版本，然后将此输出舍入为硬聚类。我们提供了一个通用的方法来证明该算法的性能保证，我们分析了该算法的背景下，亚高斯混合模型。我们还研究了估计高斯中心的基本限制，以比较我们的近似保证理论上最优的k-均值聚类解决方案。

Sketch-and-solve approaches to k-means clustering by semidefinite programming

We introduce a sketch-and-solve approach to speed up the Peng-Wei semidefinite relaxation of k-means clustering. When the data is appropriately separated we identify the k-means optimal clustering. Otherwise, our approach provides a high-confidence lower bound on the optimal k-means value. This lower bound is data-driven; it does not make any assumption on the data nor how it is generated. We provide code and an extensive set of numerical experiments where we use this approach to certify approximate optimality of clustering solutions obtained by k-means++.