We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.

中文翻译：

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分段线性感知器中非线性激励的扩散

我们研究了连续非凸优化问题，具有分段线性成本函数的球面感知器的能量景观的局部极小值的性质，并表明它们是临界的，边缘稳定的，并显示出一组伪间隙，奇点和非线性激励其性质似乎与硬球的堵塞堆积物具有相同的普遍性。分段线性感知器问题表现为最近在[1]中研究的纯线性感知器优化问题的演变。它的成本函数包含两个非解析点，其中导数有跳跃。相应地，在非凸/玻璃态阶段，这两个点在力分布中产生了四个伪间隙，这在间隙分布中也引起了四个幂定律。另外，可以定义等静性的扩展概念，并表明在该阶段局部极小值再次显得是等静的。我们相信，随着成本函数中非分析点数量的增加，我们的结果自然地会推广到非线性激励激增的更为复杂的情况。