69 research outputs found
Neural Network Verification as Piecewise Linear Optimization: Formulations for the Composition of Staircase Functions
We present a technique for neural network verification using mixed-integer
programming (MIP) formulations. We derive a \emph{strong formulation} for each
neuron in a network using piecewise linear activation functions. Additionally,
as in general, these formulations may require an exponential number of
inequalities, we also derive a separation procedure that runs in super-linear
time in the input dimension. We first introduce and develop our technique on
the class of \emph{staircase} functions, which generalizes the ReLU, binarized,
and quantized activation functions. We then use results for staircase
activation functions to obtain a separation method for general piecewise linear
activation functions. Empirically, using our strong formulation and separation
technique, we can reduce the computational time in exact verification settings
based on MIP and improve the false negative rate for inexact verifiers relying
on the relaxation of the MIP formulation
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Discrete isometry groups of symmetric spaces
This survey is based on a series of lectures that we gave at MSRI in Spring
2015 and on a series of papers, mostly written jointly with Joan Porti. Our
goal here is to:
1. Describe a class of discrete subgroups of higher rank
semisimple Lie groups, which exhibit some "rank 1 behavior".
2. Give different characterizations of the subclass of Anosov subgroups,
which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of
various equivalent dynamical and geometric properties (such as asymptotically
embedded, RCA, Morse, URU).
3. Discuss the topological dynamics of discrete subgroups on flag
manifolds associated to and Finsler compactifications of associated
symmetric spaces . Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces .Comment: 77 page
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