PL Manifolds Research Articles

Piecewise Linear Transformation – Propagating Aleatoric Uncertainty in Neural Networks

Real-world data typically exhibit aleatoric uncertainty which has to be considered during data-driven decision-making to assess the confidence of the decision provided by machine learning models. To propagate aleatoric uncertainty represented by probability distributions (PDs) through neural networks (NNs), both sampling-based and function approximation-based methods have been proposed. However, these methods suffer from significant approximation errors and are not able to accurately represent predictive uncertainty in the NN output. In this paper, we present a novel method, Piecewise Linear Transformation (PLT), for propagating PDs through NNs with piecewise linear activation functions (e.g., ReLU NNs). PLT does not require sampling or specific assumptions about the PDs. Instead, it harnesses the piecewise linear structure of such NNs to determine the propagated PD in the output space. In this way, PLT supports the accurate quantification of predictive uncertainty based on the criterion exactness of the propagated PD. We assess this exactness in theory by showing error bounds for our propagated PD. Further, our experimental evaluation validates that PLT outperforms competing methods on publicly available real-world classification and regression datasets regarding exactness. Thus, the PDs propagated by PLT allow to assess the uncertainty of the provided decisions, offering valuable support.

Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

Abstract We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n \mathbb{Z}_{n} grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 \pi_{1} -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated.

On the relative L-theory and the relative signature of PL manifolds with boundary

In this paper, we give a new description of the relative topological structure group of PL manifolds with boundary, and put the relative L-group into an exact sequence of groups. Then we define the relative index of PL manifolds with boundary, and show that this relative index will induce an additive map from the relative L-group to the K-theory of the relative Roe algebra.

Gem-induced trisections of compact PL 4-manifolds

Abstract The idea of studying trisections of closed smooth 4-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to standard simply-connected 4-manifolds, via the so-called simple crystallizations. In the present paper we propose generalizations of these ideas by taking into consideration a possible extension of trisections to compact PL 4-manifolds with connected boundary, which is related to Birman’s special Heegaard sewing, and by analyzing gem-induced trisections, i.e. trisections that can be induced not only by simple crystallizations, but also by any 5-colored graph encoding a PL 4-manifold with empty or connected boundary. This last notion gives rise to that of G-trisection genus, as an analogue, in this context, of the well-known trisection genus. We give conditions on a 5-colored graph ensuring one of its gem-induced trisections – if any – to realize the G-trisection genus, and prove how to determine it directly from the graph. As a consequence, we detect a class of closed simply-connected 4-manifolds, comprehending all standard ones, for which both G-trisection genus and trisection genus coincide with the second Betti number and also with half the value of the graph-defined PL invariant regular genus. Moreover, the existence of gem-induced trisections and an estimation of the G-trisection genus via surgery description is obtained, for each compact PL 4-manifold admitting a handle decomposition lacking in 3-handles.

Discovering a change point and piecewise linear structure in a time series of organoid networks via the iso-mirror

Recent advancements have been made in the development of cell-based in-vitro neuronal networks, or organoids. In order to better understand the network structure of these organoids, a super-selective algorithm has been proposed for inferring the effective connectivity networks from multi-electrode array data. In this paper, we apply a novel statistical method called spectral mirror estimation to the time series of inferred effective connectivity organoid networks. This method produces a one-dimensional iso-mirror representation of the dynamics of the time series of the networks which exhibits a piecewise linear structure. A classical change point algorithm is then applied to this representation, which successfully detects a change point coinciding with the neuroscientifically significant time inhibitory neurons start appearing and the percentage of astrocytes increases dramatically. This finding demonstrates the potential utility of applying the iso-mirror dynamic structure discovery method to inferred effective connectivity time series of organoid networks.

A Piecewise Linear Regression and Classification Algorithm With Application to Learning and Model Predictive Control of Hybrid Systems

This paper proposes an algorithm for solving multivariate regression and classification problems using piecewise linear predictors over a polyhedral partition of the feature space. The resulting algorithm that we call PARC (Piecewise Affine Regression and Classification) alternates between (<i>i</i>) solving ridge regression problems for numeric targets, softmax regression problems for categorical targets, and either softmax regression or cluster centroid computation for piecewise linear separation, and (<i>ii</i>) assigning the training points to different clusters on the basis of a criterion that balances prediction accuracy and piecewise-linear separability. We prove that PARC is a block-coordinate descent algorithm that minimizes a suitably constructed objective function and that it converges in a finite number of steps. The algorithm is used to learn hybrid numerical/categorical (HYNC) dynamical models from data that contain real and discrete labeled values. The resulting model has a piecewise linear structure that is particularly useful to formulate model predictive control problems and solve them by mixed-integer programming. A Python implementation of PARC is available at <uri>http://cse.lab.imtlucca.it/~bemporad/parc</uri>.

Handle decompositions for a class of closed orientable PL 4-manifolds

In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{{Z}}$, we have constructed a handle decomposition for $M$ as one of the following types: $(1)$ one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, $(2)$ one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients.

PL Morse theory in low dimensions

Abstract We discuss a PL analog of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level; it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular, we show that in dimensions d ≤ 4 a homologically regular point on a PL d-manifold is always strongly regular. Examples show that this fails in higher dimensions d ≥ 5. One of our constructions involves an embedding of the dunce hat into 4-space and Mazur’s contractible 4-manifold. Finally, decidability questions in this context are discussed.

Classifying compact 4-manifolds via generalized regular genus and G-degree

$(d+1)$-colored graphs, i.e., edge-colored graphs that are $(d+1)$-regular, have already been proved to be a useful representation tool for compact PL $d$-manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role. The present paper focuses in particular on generalized regular genus and G-degree: the first one extending to higher dimension the classical notion of Heegaard genus for 3-manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two. We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums. We also compute both generalized regular genus and G-degree for interesting classes of compact $d$-manifolds, such as handlebodies, products of closed manifolds by the interval and $\mathbb D^2$-bundles over $\mathbb S^2.$ The main results of the paper concern dimension 4, where we obtain the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with G-degree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to G-degree 24.

Boundary measurement and sign variation in real projective space

We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space.We find the spaces to be PL manifolds with boundary which are homotopy equivalent to another real projective space of smaller dimension. One generalization makes use of sign variation while the other uses boundary measurement. Spaces arising from boundary measurement are shown to admit Cohen–Macaulay triangulations.

On a buffered threshold autoregressive stochastic volatility model

AbstractThis article introduces a new autoregressive stochastic volatility (SV) model with a new piecewise linear structure such that the regime‐switching mechanism has a buffer zone where regime‐switching is delayed. The proposed model allows us to model the hysteretic phenomenon of the regime‐switching existing on both the mean equation and the volatility equation. A full description of the proposed Markov chain Monte Carlo method is given. In the empirical study, we consider the daily closing prices of NIKKEI stock average, the exchange rate for US Dollar to Japanese Yen and Hang Seng Index. Deviance information criterion measure shows that our proposed model outperforms the classical threshold SV models.

Effective Actions for Regge Piecewise Flat Quantum Gravity

We review the construction of the path integral and the corresponding effective action for the Regge formulation of General Relativity under the assumption that the short-distance structure of the spacetime is not a smooth 4-manifold, but a piecewise linear manifold based on a triangulation of a smooth 4-manifold. We point out that the exponentially damped 4-volume path-integral measure does not give a finite path integral, although it can be used for the construction of the perturbative effective action. We modify the 4-volume measure by multiplying it by an inverse power of the product of the edge-lengths such that the new measure gives a finite path integral while it retains all the nice features of the unmodified measure.

Compact 4-manifolds admitting special handle decompositions

In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of special handle decompositions for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL 4-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants regular genus, gem-complexity or G-degree among all such manifolds with the same second Betti number.

On reduced complexity of closed piecewise linear 5-manifolds

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5-manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of $$\mathbb {S}^2 \times \mathbb {S}^3$$ among closed connected spin PL 5-manifolds.

Modal truncation damping in reduced modal analysis of piecewise linear continuum structures

Modal analysis is an often-used method for the solution of dynamical problems in engineering practice. In the case of continuous systems and discretization methods (e.g., FEM) with a large number of degrees of freedom, the reduced modal analysis is used, where only a selected number of modes are applied, while the higher modes are neglected. A piecewise linear elastic structure has state-dependent stiffness and as such, state-dependent vibration modes. During the dynamic modal analysis of such systems, one has to switch between the modes. During the switch a part of the kinetic and potential energy is transferring into modes, which are neglected in the reduced modal analysis. Thus, using the reduced modal analysis for the numerical calculation of the dynamic behavior of piecewise linear elastic structures introduces the modal truncation damping. In this paper, we present on a continuous system, how this modal truncation damping can be characterized, what are the upper and lower limits of that damping and how the characteristics of the free vibration of a piecewise linear elastic structure can be predicted.

Event-triggered L1 filtering for uncertain networked control systems with multiple sensor fault modes

Considering the influence of sensor fault modes on the system performance in communication network, under the discrete event-triggered communication scheme (DETCS), the problems of the robust [Formula: see text] filtering for a class of uncertain networked control systems (NCSs) with multiple sensor fault modes and persistent and amplitude-bounded disturbance constraints are investigated. A set of stochastic variables are adopted to describe the sensor faults, the filtering error system is established, which characterizes the effects of sensor faults and DETCS. By constructing an appropriate delay-dependent Lyapunov-Krasovskii functional, new results on stability and robust [Formula: see text] performance are proposed for the filtering error system according to Lyapunov theory and the integral inequality method, and the co-design method for gaining the desired [Formula: see text] filter parameters and event-triggering parameters is given in terms of linear matrix inequalities (LMIs). We notice that with the structrue of Lyapunov-Krasovskii functional which considers the piecewise-linear sawtooth structure characteristic of transmission delay, a less conservative result is obtained. Finally, three examples are provided to illustrate the feasibility of the proposed method.

Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary

Abstract Let 𝑀 be a connected compact PL 4-manifold with boundary. In this article, we give several lower bounds for regular genus and gem-complexity of the manifold 𝑀. In particular, we prove that if 𝑀 is a connected compact 4-manifold with ℎ boundary components, then its gem-complexity k ⁢ ( M ) k(M) satisfies the inequalities k ⁢ ( M ) ≥ 3 ⁢ χ ⁢ ( M ) + 7 ⁢ m + 7 ⁢ h - 10 k(M)\geq 3\chi(M)+7m+7h-10 and k ⁢ ( M ) ≥ k ⁢ ( ∂ ⁡ M ) + 3 ⁢ χ ⁢ ( M ) + 4 ⁢ m + 6 ⁢ h - 9 k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9 , and its regular genus G ⁢ ( M ) \mathcal{G}(M) satisfies the inequalities G ⁢ ( M ) ≥ 2 ⁢ χ ⁢ ( M ) + 3 ⁢ m + 2 ⁢ h - 4 \mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and G ⁢ ( M ) ≥ G ⁢ ( ∂ ⁡ M ) + 2 ⁢ χ ⁢ ( M ) + 2 ⁢ m + 2 ⁢ h - 4 \mathcal{G}(M)\geq\mathcal{G}(\partial M)+2\chi(M)+2m+2h-4 , where 𝑚 is the rank of the fundamental group of the manifold 𝑀. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds is also shown for a large class of PL 4-manifolds with boundary.

Co-design between robust L 1 fault-tolerant control and discrete event-triggered communication scheme for networked control systems with transmission delay and quantisation

In order to take into account both the system performance and economical utilisation of network resources, under the discrete event-triggered communication scheme, the problems of the robust fault-tolerant control for a class of uncertain networked control systems with actuator fault constraint are investigated. In consideration with the imperfect networked environment with transmission delay and quantisation, a model of the error dependent networked closed-loop fault system is established. By constructing an appropriate delay-dependent Lyapunov–Krasovskii functional, the sufficient condition for the existence of robust fault-tolerant controller is obtained for an error dependent networked closed-loop fault system, and a co-design method for gaining the discrete event-triggered parameter and fault-tolerant controller is proposed in terms of linear matrix inequalities. Notice that the structure of Lyapunov–Krasovskii functional which considers the piecewise-linear sawtooth structure characteristic of transmission delay, a less conservative result is obtained. Finally, simulation examples are provided to illustrate the feasibility of the proposed method.

NUMERICAL SCHEME OF MULTI-MATERIAL COMPRESSIBLE FLOW WITH SHARP INTERFACE ON EULERIAN GRIDS

Numerical simulation of multi-material compressible flow problem is of great importance in both the national defense and industry areas, such as weapon design and blast wave defense. Due to the property of large deformation and high nonlinearity, the efficient simulation of multi-material compressible flow becomes a quite challenging problem. A numerical scheme is developed to carry out the simulation of an immiscible multi-material compressible flow with sharp phase interface on two dimensional and three dimensional unstructured Eulerian grids, which can handle the large deformation of compressible fluid and elastoplastic solid under the extreme conditions. We use the level set method to depict the phase interface numerically, and explicitly reconstruct the phase interface in a piecewise linear manifold. The topological structure of the phase interface is constructed explicitly, which can handle any number of media in the whole computational domain and three media in a single cell. The traditional finite volume method is used to calculate the edge numeircal flux between the same material in adjacent cells, while the phase interface flux is calculated by exactly solving a one dimensional multi-material Riemann problem on the normal direction of the phase interface. The above procedures can keep the conservation of the phase interface flux, and the interaction between two media across the phase interface can keep consistent with the real situation. A robust aggregation algorithm is adopted to build cell patches and adjust the conservation variables around the phase interface, which can effectively remove the numerical instability due to the breakdown of the CFL constraint by the cell fragments. Some classical examples and application problems, such as one-dimensional multi-material Riemann problem, gas-bubble interaction problem, intensive airblast problem, sub-surface blast problem, and blast wave propagation in three dimensional sap, which have a good agreement with the corresponding analytical and experimental results, are presented to validate our numerical scheme.

On Treed Gaussian Processes and piecewise-linear NARX modelling

In the scope of nonlinear system identification, traditional parametric models are widely adopted as simplifying approaches to modelling the complexity of nonlinearity. However, many high-order parametric models are disadvantaged due to their inherent demand for model detection and their tendency to overfit in the absence of additional validation processes. Nonparametric models, such as the Gaussian Process (GP), though being naturally exempt from model detection, can involve expensive procedures of model optimisation. This article presents a Linear Kernel Chipman-based Treed Gaussian Processes (LK-CTGP), which is essentially an assembly of simple linear parametric models using a decision tree framework, to model nonlinear systems. The piecewise-linear structure of the LK-CTGP offers a natural geometric solution to modelling nonlinear systems, where no model detection is required. The essence of simplicity from the traditional parametric model is also completely retained within each of the submodels. The effectiveness of the LK-CTGP is illustrated here via a number of case studies from simple synthetic data to experimental data, on which Nonlinear Autoregressive eXogenous (NARX) systems will built from the data for in-depth study.