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Register a new userGlobally Injective ReLU Networks
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Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end -- rather than layerwise -- doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.
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It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way. Here, we prove that in fact it is often possible to identify the architecture, weights, and biases of an unknown deep ReLU network by observing only its output. Every ReLU network defines a piecewise linear function, where the boundaries between linear regions correspond to inputs for which some neuron in the network switches between inactive and active ReLU states. By dissecting the set of region boundaries into components associated with particular neurons, we show both theoretically and empirically that it is possible to recover the weights of neurons and their arrangement within the network, up to isomorphism.
We present a theoretical and empirical study of the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes:kernel and adaptive, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavior of initialization coefficients in the limit of large network widths. We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces to cubic splines for uniform initializations. Learning in the adaptive regime favors instead linear splines, where knots cluster adaptively at the sample points.
Before training a neural net, a classic rule of thumb is to randomly initialize the weights so the variance of activations is preserved across layers. This is traditionally interpreted using the total variance due to randomness in both weights emph{and} samples. Alternatively, one can interpret the rule of thumb as preservation of the variance over samples for a fixed network. The two interpretations differ little for a shallow net, but the difference is shown to grow with depth for a deep ReLU net by decomposing the total variance into the network-averaged sum of the sample variance and square of the sample mean. We demonstrate that even when the total variance is preserved, the sample variance decays in the later layers through an analytical calculation in the limit of infinite network width, and numerical simulations for finite width. We show that Batch Normalization eliminates this decay and provide empirical evidence that preserving the sample variance instead of only the total variance at initialization time can have an impact on the training dynamics of a deep network.
Recently, path norm was proposed as a new capacity measure for neural networks with Rectified Linear Unit (ReLU) activation function, which takes the rescaling-invariant property of ReLU into account. It has been shown that the generalization error bound in terms of the path norm explains the empirical generalization behaviors of the ReLU neural networks better than that of other capacity measures. Moreover, optimization algorithms which take path norm as the regularization term to the loss function, like Path-SGD, have been shown to achieve better generalization performance. However, the path norm counts the values of all paths, and hence the capacity measure based on path norm could be improperly influenced by the dependency among different paths. It is also known that each path of a ReLU network can be represented by a small group of linearly independent basis paths with multiplication and division operation, which indicates that the generalization behavior of the network only depends on only a few basis paths. Motivated by this, we propose a new norm emph{Basis-path Norm} based on a group of linearly independent paths to measure the capacity of neural networks more accurately. We establish a generalization error bound based on this basis path norm, and show it explains the generalization behaviors of ReLU networks more accurately than previous capacity measures via extensive experiments. In addition, we develop optimization algorithms which minimize the empirical risk regularized by the basis-path norm. Our experiments on benchmark datasets demonstrate that the proposed regularization method achieves clearly better performance on the test set than the previous regularization approaches.
We explore convergence of deep neural networks with the popular ReLU activation function, as the depth of the networks tends to infinity. To this end, we introduce the notion of activation domains and activation matrices of a ReLU network. By replacing applications of the ReLU activation function by multiplications with activation matrices on activation domains, we obtain an explicit expression of the ReLU network. We then identify the convergence of the ReLU networks as convergence of a class of infinite products of matrices. Sufficient and necessary conditions for convergence of these infinite products of matrices are studied. As a result, we establish necessary conditions for ReLU networks to converge that the sequence of weight matrices converges to the identity matrix and the sequence of the bias vectors converges to zero as the depth of ReLU networks increases to infinity. Moreover, we obtain sufficient conditions in terms of the weight matrices and bias vectors at hidden layers for pointwise convergence of deep ReLU networks. These results provide mathematical insights to the design strategy of the well-known deep residual networks in image classification.
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