We introduce squared neural Poisson point processes (SNEPPPs) by parameterising the intensity function by the squared norm of a two layer neural network. When the hidden layer is fixed and the second layer has a single neuron, our approach resembles previous uses of squared Gaussian process or kernel methods, but allowing the hidden layer to be learnt allows for additional flexibility. In many cases of interest, the integrated intensity function admits a closed form and can be computed in quadratic time in the number of hidden neurons. We enumerate a far more extensive number of such cases than has previously been discussed. Our approach is more memory and time efficient than naive implementations of squared or exponentiated kernel methods or Gaussian processes. Maximum likelihood and maximum a posteriori estimates in a reparameterisation of the final layer of the intensity function can be obtained by solving a (strongly) convex optimisation problem using projected gradient descent. We demonstrate SNEPPPs on real, and synthetic benchmarks, and provide a software implementation. https://github.com/RussellTsuchida/snefy

翻译：我们通过将强度函数参数化为两层神经网络范数的平方，引入了平方神经泊松点过程（SNEPPPs）。当隐藏层固定且第二层仅包含单个神经元时，我们的方法与先前使用平方高斯过程或核方法的方式类似，但允许学习隐藏层则提供了额外的灵活性。在许多感兴趣的案例中，积分强度函数具有闭式解，并且可以在隐藏层神经元数量的二次时间内计算。我们列举了比先前讨论更为广泛的此类案例。与平方或指数核方法或高斯过程的朴素实现相比，我们的方法在内存和时间效率上更优。通过使用投影梯度下降求解（严格）凸优化问题，可以获得强度函数最后一层重新参数化后的最大似然和最大后验估计。我们在真实与合成基准上验证了SNEPPPs，并提供了软件实现。https://github.com/RussellTsuchida/snefy