最简单的干扰模式
The simplest model of jamming
摘要：
我们研究了一个众所周知的机器学习模型-感知器-作为一个简单的模型干扰的硬对象。我们展示了两个制度：1）凸优化制度，其中干扰是实体的和非关键的。2)非凸优化机制，其中干扰是均衡的和关键的。通过描述力和间隙分布规律的指数来描述临界干扰阶段。令人惊讶的是，我们发现，这些指数与最近在高维硬球计算的相应的。此外，修改感知器的随机线性规划问题，我们证明，均衡是不是一个充分条件奇异力和间隙分布。因此，解空间的分裂（复本对称性破缺）似乎是一个关键因素。我们假设的普遍性为一大类非凸约束满足问题的连续变量。
英文摘要：
We study a well known machine learning model -the perceptron- as a simple model of jamming of hard objects. We exhibit two regimes: 1) a convex optimisation regime where jamming is hypostatic and non-critical. 2) a non convex optimisation regime where jamming is isostatic and critical. We characterise the critical jamming phase through exponents describing the distributions law of forces and gaps. Surprisingly we find that these exponents coincide with the corresponding ones recently computed in high dimensional hard spheres. In addition, modifying the perceptron to a random linear programming problem, we show that isostaticity is not a sufficient condition for singular force and gap distributions. For that, fragmentation of the space of solutions (replica symmetry breaking) appears to be a crucial ingredient. We hypothesise universality for a large class of non-convex constrained satisfaction problems with continuous variables.
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