Numerical Verification of Affine Systems with up to a Billion Dimensions

Affine systems reachability is the basis of many verification methods. With further computation, methods exist to reason about richer models with inputs, nonlinear differential equations, and hybrid dynamics. As such, the scalability of affine systems verification is a prerequisite to scalable analysis for more complex systems. In this paper, we improve the scalability of affine systems verification, in terms of the number of dimensions (variables) in the system. The reachable states of affine systems can be written in terms of the matrix exponential, and safety checking can be performed at specific time steps with linear programming. Unfortunately, for large systems with many state variables, this direct approach requires an intractable amount of memory while using an intractable amount of computation time. We overcome these challenges by combining several methods that leverage common problem structure. Memory is reduced by exploiting initial states that are not full-dimensional and safety properties (outputs) over a few linear projections of the state variables. Computation time is saved by using numerical simulations to compute only projections of the matrix exponential relevant for the verification problem. Since large systems often have sparse dynamics, we use Krylov-subspace simulation approaches based on the Arnoldi or Lanczos iterations. Our method produces accurate counter-examples when properties are violated and, in the extreme case with sufficient problem structure, can analyze a system with one billion real-valued state variables

Energy-momentum diffusion from spacetime discreteness

We study potentially observable consequences of spatiotemporal discreteness for the motion of massive and massless particles. First we describe some simple intrinsic models for the motion of a massive point particle in a fixed causal set background. At large scales, the microscopic swerves induced by the underlying atomicity manifest themselves as a Lorentz invariant diffusion in energy-momentum governed by a single phenomenological parameter, and we derive in full the corresponding diffusion equation. Inspired by the simplicity of the result, we then derive the most general Lorentz invariant diffusion equation for a massless particle, which turns out to contain two phenomenological parameters describing, respectively, diffusion and drift in the particle's energy. The particles do not leave the light cone however: their worldlines continue to be null geodesics. Finally, we deduce bounds on the drift and diffusion constants for photons from the blackbody nature of the spectrum of the cosmic microwave background radiation.Comment: 13 pages, 4 figures, corrected minor typos and updated to match published versio

Deep Self-Taught Learning for Handwritten Character Recognition

Recent theoretical and empirical work in statistical machine learning has demonstrated the importance of learning algorithms for deep architectures, i.e., function classes obtained by composing multiple non-linear transformations. Self-taught learning (exploiting unlabeled examples or examples from other distributions) has already been applied to deep learners, but mostly to show the advantage of unlabeled examples. Here we explore the advantage brought by {\em out-of-distribution examples}. For this purpose we developed a powerful generator of stochastic variations and noise processes for character images, including not only affine transformations but also slant, local elastic deformations, changes in thickness, background images, grey level changes, contrast, occlusion, and various types of noise. The out-of-distribution examples are obtained from these highly distorted images or by including examples of object classes different from those in the target test set. We show that {\em deep learners benefit more from out-of-distribution examples than a corresponding shallow learner}, at least in the area of handwritten character recognition. In fact, we show that they beat previously published results and reach human-level performance on both handwritten digit classification and 62-class handwritten character recognition

Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets

Reachability analysis is a formal method to guarantee safety of dynamical systems under the influence of uncertainties. A major bottleneck of all reachability algorithms is the requirement to adequately tune certain algorithm parameters such as the time step size, which requires expert knowledge. In this work, we solve this issue with a fully-automated reachability algorithm that tunes all algorithm parameters internally such that the reachable set enclosure satisfies a user-defined accuracy in terms of distance to the exact reachable set. Knowing the distance to the exact reachable set, an inner-approximation of the reachable set can be efficiently extracted from the outer-approximation using the Minkowski difference. Finally, we propose a novel verification algorithm that automatically refines the accuracy of the outer- and inner-approximation until specifications given by time-varying safe and unsafe sets can either be verified or falsified. The numerical evaluation demonstrates that our verification algorithm successfully verifies or falsifies benchmarks from different domains without any requirement for manual tuning.Comment: 16 page