Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with non-linear excitations [1, 2]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to ‘surf’ between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior.

最近，我们发现，对于连续变量和软排除的体积约束，无论是无限尺寸还是有限尺寸的优化问题，都可以显示整个均衡阶段，其中成本函数的局部最小值是具有非线性激励的边际稳定配置[ 1、2]。在这项工作中，我们描述了一种绝热绝热算法，以连续地探索相应的粗糙高维景观。我们专注于这类原型问题，即具有线性成本函数（铰链损耗）的球形感知器优化问题。该算法允许在等静压的边缘稳定配置之间“冲浪”，并研究此类景观的某些属性。特别是，我们专注于局部极小值不稳定时发生的雪崩的统计数据。我们表明，当扰动这样的最小值时，系统会发生可塑性重排，其大小是幂律分布的，并且我们表征了相应的临界指数。最后，我们研究了无干扰跃迁的关键特性，表明线性相互作用势在能量和压力的比例关系中产生了对数行为，该对数行为是距无干扰点的距离的函数。对于某些数量，可以计算出对数校正。违反软约束的数量就是这种情况，软约束的数量取决于到遵循非平凡幂律行为的干扰距离。最后，我们研究了无干扰跃迁的关键特性，表明线性相互作用势在能量和压力的比例关系中产生了对数行为，该对数行为是距无干扰点的距离的函数。对于某些数量，可以计算出对数校正。违反软约束的数量就是这种情况，软约束的数量取决于到遵循非平凡幂律行为的干扰距离。最后，我们研究了无干扰跃迁的关键特性，表明线性相互作用势在能量和压力的比例关系中产生了对数行为，该对数行为是距无干扰点的距离的函数。对于某些数量，可以计算出对数校正。违反软约束的数量就是这种情况，软约束的数量取决于到遵循非平凡幂律行为的干扰距离。