生活在边缘：随机数据凸规划中的相变
Living on the edge: Phase transitions in convex programs with random data
摘要：
最近的研究表明，许多随机约束的凸优化问题表现出相变的约束数量的增加。例如，这种现象出现在用于从随机线性测量识别稀疏向量的$\ell_1$最小化方法中。实际上，当测量的数量超过取决于稀疏水平的阈值时，$\ell_1$方法以高概率成功;否则，它以高概率失败。本文提供了第一个严格的分析，解释了为什么相变是无处不在的随机凸优化问题。它还描述了对过渡的定量方面进行可靠预测的工具，包括过渡区的位置和宽度。这些技术适用于正则化的线性逆问题与随机测量，随机不相干模型下的分层问题，也随机仿射约束锥程序。应用结果依赖于圆锥曲线几何的基础研究。本文引入了一个概括参数，称为统计维数，规范地扩展了一个线性子空间的凸锥类的维数。主要技术结果表明，凸锥的内禀体积序列集中在统计维数附近。这一事实导致了一个随机旋转的圆锥体与一个固定的圆锥体共享一条射线的概率的精确界限。
英文摘要：
Recent research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the $\ell_1$ minimization method for identifying a sparse vector from random linear measurements. Indeed, the $\ell_1$ approach succeeds with high probability when the number of measurements exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability. This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints. The applied results depend on foundational research in conic geometry. This paper introduces a summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. The main technical result demonstrates that the sequence of intrinsic volumes of a convex cone concentrates sharply around the statistical dimension. This fact leads to accurate bounds on the probability that a randomly rotated cone shares a ray with a fixed cone.
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