论文标题
深层神经网络的局部识别性：理论
Local Identifiability of Deep ReLU Neural Networks: the Theory
论文作者
论文摘要
样本是否足够丰富，至少在本地确定神经网络的参数？为了回答这个问题，我们通过固定其某些权重的值引入了给定深层神经网络的新局部参数化。这使我们能够定义本地提升操作员，其倒置是高维空间的平滑歧管的图表。 Deep Relu神经网络实现的函数由依赖样本的线性操作员组成局部提升。我们从这种方便的表示中得出了局部可识别性的几何必要条件。查看切线空间，几何条件提供了：1/可识别性的尖锐而可测试的必要条件，以及2/可识别局部可识别性的尖锐且可测试的足够条件。可以使用反向传播和矩阵等级计算对条件的有效性进行数值测试。
Is a sample rich enough to determine, at least locally, the parameters of a neural network? To answer this question, we introduce a new local parameterization of a given deep ReLU neural network by fixing the values of some of its weights. This allows us to define local lifting operators whose inverses are charts of a smooth manifold of a high dimensional space. The function implemented by the deep ReLU neural network composes the local lifting with a linear operator which depends on the sample. We derive from this convenient representation a geometrical necessary and sufficient condition of local identifiability. Looking at tangent spaces, the geometrical condition provides: 1/ a sharp and testable necessary condition of identifiability and 2/ a sharp and testable sufficient condition of local identifiability. The validity of the conditions can be tested numerically using backpropagation and matrix rank computations.
