Empirical works show that for ReLU neural networks (NNs) with small initialization, input weights of hidden neurons (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) condense onto isolated orientations. The condensation dynamics implies that the training implicitly regularizes a NN towards one with much smaller effective size. In this work, we illustrate the formation of the condensation in multi-layer fully connected NNs and show that the maximal number of condensed orientations in the initial training stage is twice the multiplicity of the activation function, where "multiplicity" indicates the multiple roots of activation function at origin. Our theoretical analysis confirms experiments for two cases, one is for the activation function of multiplicity one with arbitrary dimension input, which contains many common activation functions, and the other is for the layer with one-dimensional input and arbitrary multiplicity. This work makes a step towards understanding how small initialization leads NNs to condensation at the initial training stage.

Empirical works show that for ReLU neural networks (NNs) with small initialization, input weights of hidden neurons (the input weight of a hidden neuron consists oftheweight fromitsinputlayertothehidden neuron anditsbiasterm) condense ontoisolated orientations.

Motivated by a recent result of Daskalakis et al. \cite{DGTZ18}, we analyze the population version of Expectation-Maximization (EM) algorithm for the case of \textit{truncated} mixtures of two Gaussians. Truncated samples from a $d$-dimensional mixture of two Gaussians $\frac{1}{2} \mathcal{N}(\vec{\mu}, \vec{\Sigma})+ \frac{1}{2} \mathcal{N}(-\vec{\mu}, \vec{\Sigma})$ means that a sample is only revealed if it falls in some subset $S \subset \mathbb{R}^d$ of positive (Lebesgue) measure. We show that for $d=1$, EM converges almost surely (under random initialization) to the true mean (variance $\sigma^2$ is known) for any measurable set $S$. Moreover, for $d>1$ we show EM almost surely converges to the true mean $\vec{\mu}$ for any measurable set $S$, under the assumption that the map of EM has only three fixed points, namely $-\vec{\mu}, \vec{0}, \vec{\mu}$ (covariance matrix $\vec{\Sigma}$ is known). Our techniques deviate from those of Daskalakis et al. \cite{DTZ17}, which heavily depend on symmetry that the untruncated problem exhibits. The fact that the truncated set $S$ may not be symmetric around $\vec{0}$ makes it impossible to compute a closed form of the update rule of EM. We circumvent this fact by using techniques from dynamical systems, in particular implicit function theorem and stability analysis around the fixed points of the update rule of EM.