Piecewise linear manifold
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A Unified View of Piecewise Linear Neural Network Verification Open
The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. Despite the reputation of learned NN models to behave as black boxes and t…
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Branch and Bound for Piecewise Linear Neural Network Verification Open
The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. In this context, verification involves proving or disproving that an NN mo…
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Limit cycles created by piecewise linear centers Open
In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles pl…
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Efficient Representation of Low-Dimensional Manifolds using Deep Networks Open
We consider the ability of deep neural networks to represent data that lies near a low-dimensional manifold in a high-dimensional space. We show that deep networks can efficiently extract the intrinsic, low-dimensional coordinates of such …
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Transforming the canonical piecewise-linear model into a smooth-piecewise representation Open
A smoothed representation (based on natural exponential and logarithmic functions) for the canonical piecewise-linear model, is presented. The result is a completely differentiable formulation that exhibits interesting properties, like pre…
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Stability analysis of piecewise affine discrete-time systems Open
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Towards Robust, Locally Linear Deep Networks Open
Deep networks realize complex mappings that are often understood by their locally linear behavior at or around points of interest. For example, we use the derivative of the mapping with respect to its inputs for sensitivity analysis, or to…
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Limit cycles in Filippov systems having a circle as switching manifold Open
It is known that planar discontinuous piecewise linear differential systems separated by a straight line have no limit cycles when both linear differential systems are centers. Here, we study the limit cycles of the planar discontinuous pi…
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Commensurating actions for groups of piecewise continuous transformations Open
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating …
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The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line Open
The study of the piecewise linear differential systems goes back to Andronov, Vitt and Khaikin in 1920’s, and nowadays such systems still continue to receive the attention of many researchers mainly due to their applications. We study the …
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Efficient Representation of Low-Dimensional Manifolds using Deep\n Networks Open
We consider the ability of deep neural networks to represent data that lies\nnear a low-dimensional manifold in a high-dimensional space. We show that deep\nnetworks can efficiently extract the intrinsic, low-dimensional coordinates of\nsu…
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Robust computation of implicit surface networks for piecewise linear functions Open
Implicit surface networks, such as arrangements of implicit surfaces and materials interfaces, are used for modeling piecewise smooth or partitioned shapes. However, accurate and numerically robust algorithms for discretizing either struct…
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Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VI. The Curious Case of Two-Sided Discontinuous Functions Open
We construct a two-sided discontinuous piecewise linear minimal valid function for the 1-row Gomory--Johnson model which is not extreme, but which is not a convex combination of other piecewise linear minimal valid functions. The new funct…
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Head Pose Estimation via Manifold Learning Open
For the last decades, manifold learning has shown its advantage of efficient non-linear dimensionality reduction in data analysis. Based on the assumption that informative and discriminative representation of the data lies on a low-dimensi…
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Piecewise linear approximation for MILP leveraging piecewise convexity to improve performance Open
To realize adaptive operation planning with MILP unit commitment, piecewise-linear approximations of the functions that describe the operating behavior of devices in the energy system have to be computed. We present an algorithm to compute…
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Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds Open
We show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold sat…
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Multisections of piecewise linear manifolds Open
Recently Gay and Kirby described a new decomposition of smooth closed $4$-manifolds called a trisection. This paper generalises Heegaard splittings of $3$-manifolds and trisections of $4$-manifolds to all dimensions, using triangulations a…
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Polyhedral DC Decomposition and DCA Optimization of Piecewise Linear Functions Open
For piecewise linear functions f : R n ↦ R we show how their abs-linear representation can be extended to yield simultaneously their decomposition into a convex f ˇ and a concave part f ^ , including a pair of generalized gradients g ˇ ∈ R…
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Geodesic ray transform with matrix weights for piecewise constant functions Open
We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension thre…
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Symmetrical Piecewise Linear Functions Composed by Absolute Value Function Open
Summary We continue the formal development of the application of piecewise linear functions and centroids in the area of fuzzy set theory. The corresponding piecewise linear functions are symmetrical and composed by absolute function. In t…
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Dynamics of piecewise translation maps Open
In this paper, we introduce a generalized piecewise translation map on the Euclidean space. We provide a special case when this map is always of finite type. For a finite type map in this case, we form conjectures on the semi-continuity of…
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A Deep-Network Piecewise Linear Approximation Formula Open
The mathematical foundation of deep learning is the theorem that any continuous function can be approximated within any specified accuracy by using a neural network with certain non-linear activation functions. However, this theorem does n…
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Some classes of sets of structures definable without quantifiers Open
We derive abstract characterizations of the Strictly Piecewise Local (SPL) and Piecewise Locally Testable (PLT) stringsets.These generalize both the Strictly Local/Locally Testable stringsets (SL and LT) and Strictly Piecewise/Piecewise Te…
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On rigorous integration of continuous piecewise linear systems Open
In this work, rigorous methods to compute solutions of continuous piecewise linear systems are presented. A general technique, based on integrating a perturbed linear system, is described to compute enclosures of trajectories crossing plan…
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Piecewise Linear Units Improve Deep Neural Networks Open
The activation function is at the heart of a deep neural networks nonlinearity; the choice of the function has great impact on the success of training. Currently, many practitioners prefer the Rectified Linear Unit (ReLU) due to its simpli…
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Computing the Distance between Piecewise-Linear Bivariate Functions Open
We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M , induced by the L 2 norm—that is, ‖ f - g ‖2 = √∫ M ( f - g ) 2 . If f is defined by linear interpol…
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Equivariant smoothing of piecewise linear manifolds Open
We prove that every piecewise linear manifold of dimension up to four on which a finite group acts by piecewise linear homeomorphisms admits a compatible smooth structure with respect to which the group acts smoothly.
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Local inversion of a class of piecewise regular maps Open
This paper provides sufficient conditions for any map $L$, that is strongly piecewise linear relatively to a decomposition of $R^k$ in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at…
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An Effective Manifold Learning Approach to Parametrize Data for Generative Modeling of Biosignals Open
Modeling data generated by physiological systems is a crucial step in many problems such as classification, signal reconstruction and data augmentation. However finding appropriate models from high-dimensional data sampled from biosignals …
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A constructive enclosure approximation of a continuous function of many variables by piecewise linear functions Open
In this paper, for contributing to solve the problems of one-sided, or two-sided approximation, a constructive enclosure approximation (having representable lower and upper bounds) of a continuous function of many variables by piecewise li…