34,283 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
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Sparse Representation of Astronomical Images
Sparse representation of astronomical images is discussed. It is shown that a
significant gain in sparsity is achieved when particular mixed dictionaries are
used for approximating these types of images with greedy selection strategies.
Experiments are conducted to confirm: i)Effectiveness at producing sparse
representations. ii)Competitiveness, with respect to the time required to
process large images.The latter is a consequence of the suitability of the
proposed dictionaries for approximating images in partitions of small
blocks.This feature makes it possible to apply the effective greedy selection
technique Orthogonal Matching Pursuit, up to some block size. For blocks
exceeding that size a refinement of the original Matching Pursuit approach is
considered. The resulting method is termed Self Projected Matching Pursuit,
because is shown to be effective for implementing, via Matching Pursuit itself,
the optional back-projection intermediate steps in that approach.Comment: Software to implement the approach is available on
http://www.nonlinear-approx.info/examples/node1.htm
The curvelet transform for image denoising
We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement
Family of 2-simplex cognitive tools and their application for decision-making and its justifications
Urgency of application and development of cognitive graphic tools for usage
in intelligent systems of data analysis, decision making and its justifications
is given. Cognitive graphic tool "2-simplex prism" and examples of its usage
are presented. Specificity of program realization of cognitive graphics tools
invariant to problem areas is described. Most significant results are given and
discussed. Future investigations are connected with usage of new approach to
rendering, cross-platform realization, cognitive features improving and
expanding of n-simplex family.Comment: 14 pages, 6 figures, conferenc
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Formal Verification of Input-Output Mappings of Tree Ensembles
Recent advances in machine learning and artificial intelligence are now being
considered in safety-critical autonomous systems where software defects may
cause severe harm to humans and the environment. Design organizations in these
domains are currently unable to provide convincing arguments that their systems
are safe to operate when machine learning algorithms are used to implement
their software.
In this paper, we present an efficient method to extract equivalence classes
from decision trees and tree ensembles, and to formally verify that their
input-output mappings comply with requirements. The idea is that, given that
safety requirements can be traced to desirable properties on system
input-output patterns, we can use positive verification outcomes in safety
arguments. This paper presents the implementation of the method in the tool
VoTE (Verifier of Tree Ensembles), and evaluates its scalability on two case
studies presented in current literature.
We demonstrate that our method is practical for tree ensembles trained on
low-dimensional data with up to 25 decision trees and tree depths of up to 20.
Our work also studies the limitations of the method with high-dimensional data
and preliminarily investigates the trade-off between large number of trees and
time taken for verification
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