On the sensitivity of pose estimation neural networks: rotation parameterizations, Lipschitz constants, and provable bounds

In this paper, we approach the task of determining sensitivity bounds for pose estimation neural networks. This task is particularly challenging as it requires characterizing the sensitivity of 3D rotations. We develop a sensitivity measure that describes the maximum rotational change in a network's output with respect to a Euclidean change in its input. We show that this measure is a type of Lipschitz constant, and that it is bounded by the product of a network's Euclidean Lipschitz constant and an intrinsic property of a rotation parameterization which we call the "distance ratio constant". We derive the distance ratio constant for several rotation parameterizations, and then discuss why the structure of most of these parameterizations makes it difficult to construct a pose estimation network with provable sensitivity bounds. However, we show that sensitivity bounds can be computed for networks which parameterize rotation using unconstrained exponential coordinates. We then construct and train such a network and compute sensitivity bounds for it.

Analytical bounds on the local Lipschitz constants of ReLU networks

In this paper, we determine analytical upper bounds on the local Lipschitz constants of feedforward neural networks with ReLU activation functions. We do so by deriving Lipschitz constants and bounds for ReLU, affine-ReLU, and max pooling functions, and combining the results to determine a network-wide bound. Our method uses several insights to obtain tight bounds, such as keeping track of the zero elements of each layer, and analyzing the composition of affine and ReLU functions. Furthermore, we employ a careful computational approach which allows us to apply our method to large networks such as AlexNet and VGG-16. We present several examples using different networks, which show how our local Lipschitz bounds are tighter than the global Lipschitz bounds. We also show how our method can be applied to provide adversarial bounds for classification networks. These results show that our method produces the largest known bounds on minimum adversarial perturbations for large networks such as AlexNet and VGG-16.

Analytical bounds on the local Lipschitz constants of affine-ReLU functions

In this paper, we determine analytical bounds on the local Lipschitz constants of of affine functions composed with rectified linear units (ReLUs). Affine-ReLU functions represent a widely used layer in deep neural networks, due to the fact that convolution, fully-connected, and normalization functions are all affine, and are often followed by a ReLU activation function. Using an analytical approach, we mathematically determine upper bounds on the local Lipschitz constant of an affine-ReLU function, show how these bounds can be combined to determine a bound on an entire network, and discuss how the bounds can be efficiently computed, even for larger layers and networks. We show several examples by applying our results to AlexNet, as well as several smaller networks based on the MNIST and CIFAR-10 datasets. The results show that our method produces tighter bounds than the standard conservative bound (i.e. the product of the spectral norms of the layers' linear matrices), especially for small perturbations.