3,787 research outputs found
Formal Verification of Neural Network Controlled Autonomous Systems
In this paper, we consider the problem of formally verifying the safety of an
autonomous robot equipped with a Neural Network (NN) controller that processes
LiDAR images to produce control actions. Given a workspace that is
characterized by a set of polytopic obstacles, our objective is to compute the
set of safe initial conditions such that a robot trajectory starting from these
initial conditions is guaranteed to avoid the obstacles. Our approach is to
construct a finite state abstraction of the system and use standard
reachability analysis over the finite state abstraction to compute the set of
the safe initial states. The first technical problem in computing the finite
state abstraction is to mathematically model the imaging function that maps the
robot position to the LiDAR image. To that end, we introduce the notion of
imaging-adapted sets as partitions of the workspace in which the imaging
function is guaranteed to be affine. We develop a polynomial-time algorithm to
partition the workspace into imaging-adapted sets along with computing the
corresponding affine imaging functions. Given this workspace partitioning, a
discrete-time linear dynamics of the robot, and a pre-trained NN controller
with Rectified Linear Unit (ReLU) nonlinearity, the second technical challenge
is to analyze the behavior of the neural network. To that end, we utilize a
Satisfiability Modulo Convex (SMC) encoding to enumerate all the possible
segments of different ReLUs. SMC solvers then use a Boolean satisfiability
solver and a convex programming solver and decompose the problem into smaller
subproblems. To accelerate this process, we develop a pre-processing algorithm
that could rapidly prune the space feasible ReLU segments. Finally, we
demonstrate the efficiency of the proposed algorithms using numerical
simulations with increasing complexity of the neural network controller
Classification and Geometry of General Perceptual Manifolds
Perceptual manifolds arise when a neural population responds to an ensemble
of sensory signals associated with different physical features (e.g.,
orientation, pose, scale, location, and intensity) of the same perceptual
object. Object recognition and discrimination requires classifying the
manifolds in a manner that is insensitive to variability within a manifold. How
neuronal systems give rise to invariant object classification and recognition
is a fundamental problem in brain theory as well as in machine learning. Here
we study the ability of a readout network to classify objects from their
perceptual manifold representations. We develop a statistical mechanical theory
for the linear classification of manifolds with arbitrary geometry revealing a
remarkable relation to the mathematics of conic decomposition. Novel
geometrical measures of manifold radius and manifold dimension are introduced
which can explain the classification capacity for manifolds of various
geometries. The general theory is demonstrated on a number of representative
manifolds, including L2 ellipsoids prototypical of strictly convex manifolds,
L1 balls representing polytopes consisting of finite sample points, and
orientation manifolds which arise from neurons tuned to respond to a continuous
angle variable, such as object orientation. The effects of label sparsity on
the classification capacity of manifolds are elucidated, revealing a scaling
relation between label sparsity and manifold radius. Theoretical predictions
are corroborated by numerical simulations using recently developed algorithms
to compute maximum margin solutions for manifold dichotomies. Our theory and
its extensions provide a powerful and rich framework for applying statistical
mechanics of linear classification to data arising from neuronal responses to
object stimuli, as well as to artificial deep networks trained for object
recognition tasks.Comment: 24 pages, 12 figures, Supplementary Material
The Augmented Synthetic Control Method
The synthetic control method (SCM) is a popular approach for estimating the
impact of a treatment on a single unit in panel data settings. The "synthetic
control" is a weighted average of control units that balances the treated
unit's pre-treatment outcomes as closely as possible. A critical feature of the
original proposal is to use SCM only when the fit on pre-treatment outcomes is
excellent. We propose Augmented SCM as an extension of SCM to settings where
such pre-treatment fit is infeasible. Analogous to bias correction for inexact
matching, Augmented SCM uses an outcome model to estimate the bias due to
imperfect pre-treatment fit and then de-biases the original SCM estimate. Our
main proposal, which uses ridge regression as the outcome model, directly
controls pre-treatment fit while minimizing extrapolation from the convex hull.
This estimator can also be expressed as a solution to a modified synthetic
controls problem that allows negative weights on some donor units. We bound the
estimation error of this approach under different data generating processes,
including a linear factor model, and show how regularization helps to avoid
over-fitting to noise. We demonstrate gains from Augmented SCM with extensive
simulation studies and apply this framework to estimate the impact of the 2012
Kansas tax cuts on economic growth. We implement the proposed method in the new
augsynth R package
- …