arXiv subjects

Stephane Gaubert

Publications and source records attributed to Stephane Gaubert.

At least 19 recordsLinked to original sources

arXiv2026

Tropical low-rank approximation and application to optimal control of N-body systems

We study the approximation of the value function of deterministic optimal control problems with fixed initial state, motivated by $N$-body systems. In this setting, the action functional consists of local kinetic and potential terms, along with an interaction potential. We exploit this structure to approximate the value function using a tropical tensor of small rank, i.e.\ a supremum of a small number of additively separable functions. We propose a trajectory-based tropical low-rank approximation method. Rather than propagating basis functions globally, as in usual tropical numerical methods, the approximation of the value function is improved only along a sequence of relevant trajectories. The resulting approximations form a monotone family of computable lower bounds for the exact value function, with the tropical tensor rank increasing at most linearly with the number of outer iterations. Under suitable regularity assumptions, we show that at the initial state, and also at the optimal trajectory starting from this state, the lower bounds converge to the exact value. In the $N$-body setting, the generated basis functions remain additively separable across subsystems, thereby yielding a structured tropical low-rank approximation. Numerical experiments on $N$-body systems with Coulomb-type repulsion illustrate the effectiveness of the approach up to state dimension $200$, within a half hour time budget.

math.OC↗

arXiv2025

Relu and softplus neural nets as zero-sum turn-based games

We show that the output of a ReLU neural network can be interpreted as the value of a zero-sum, turn-based, stopping game, which we call the ReLU net game. The game runs in the direction opposite to that of the network, and the input of the network serves as the terminal reward of the game. In fact, evaluating the network is the same as running the Shapley-Bellman backward recursion for the value of the game. Using the expression of the value of the game as an expected total payoff with respect to the path measure induced by the transition probabilities and a pair of optimal policies, we derive a discrete Feynman-Kac-type path-integral formula for the network output. This game-theoretic representation can be used to derive bounds on the output from bounds on the input, leveraging the monotonicity of Shapley operators, and to verify robustness properties using policies as certificates. Moreover, training the neural network becomes an inverse game problem: given pairs of terminal rewards and corresponding values, one seeks transition probabilities and rewards of a game that reproduces them. Finally, we show that a similar approach applies to neural networks with Softplus activation functions, where the ReLU net game is replaced by its entropic regularization.

cs.LG↗

arXiv2024

Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and Valued Ordered fields

We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the eigenvalues and eigenvectors of these matrices. We prove that the eigenvalues of a positive (semi)-definite matrix in the tropical symmetrized setting coincide with its diagonal entries. Then, we show that the images by the valuation of the eigenvalues of a positive definite matrix over a valued nonarchimedean ordered field coincide with the eigenvalues of an associated matrix in the symmetrized tropical algebra. Moreover, under a genericity condition, we characterize the images of the eigenvectors under the map keeping track both of the nonarchimedean valuation and sign, showing that they coincide with tropical eigenvectors in the symmetrized algebra. These results offer new insights into the spectral theory of matrices over tropical semirings, and provide combinatorial formul\ae\ for log-limits of eigenvalues and eigenvectors of parametric families of real positive definite matrices.

math.RA↗

arXiv2024

Stationary regimes of piecewise linear dynamical systems with priorities

Dynamical systems governed by priority rules appear in the modeling of emergency organizations and road traffic. These systems can be modeled by piecewise linear time-delay dynamics, specifically using Petri nets with priority rules. A central question is to show the existence of stationary regimes (i.e., steady state solutions) -- taking the form of invariant half-lines -- from which essential performance indicators like the throughput and congestion phases can be derived. Our primary result proves the existence of stationary solutions under structural conditions involving the spectrum of the linear parts within the piecewise linear dynamics. This extends to a broader class of systems a fundamental theorem of Kohlberg (1980) dealing with nonexpansive dynamics. The proof of our result relies on topological degree theory and the notion of ``Blackwell optimality'' from the theory of Markov decision processes. Finally, we validate our findings by demonstrating that these structural conditions hold for a wide range of dynamics, especially those stemming from Petri nets with priority rules. This is illustrated on real-world examples from road traffic management and emergency call center operations.

math.OC↗

arXiv2023

Linear algebra over T-pairs

This is part of an ongoing project to find a general algebraic framework for semiring theory. The structure theory of semirings is quite challenging, largely because of the lack of negation, and such basic properties such as unique factorization of polynomials, multiplicativity of determinants, and the characteristic polynomial of a matrix, all fail. (In fact in the max-plus algebra, the sum of two nonzero elements is never zero!) Consequently 0 is replaced by a distinguished T-submodule $ A_0$ of $ A,$ and $( A, A_0)$ is called a ``pair.'' This paper treats linear algebra over a (not necessarily distributive) semiring pair, with a range of applications to tropical algebra as well as related areas such as hyperrings and fuzzy rings. We turn to matrices and the question of whether the row rank, column rank, and submatrix rank of a matrix are equal. The submatrix rank is less than or equal to the row rank and the column rank in many cases, including ``metatangible pairs'' with unique negation, but there is a counterexample to equality, discovered some time ago by the second author, which we provide in a more general setting (``pairs of the second kind'') that includes the hyperfield of signs. We do find situations when equality holds, encompassing results by Akian, Gaubert, Guterman, Izhakian, Knebusch, and Rowen, including versions of Cramer's rule. We pay special attention to the question of Baker and Zhang whether $n+1$ vectors of length $n$ need be dependent. At the conclusion of the main part, we consider surpassing relations, which permit us to tighten our results. The categorical setting is given in the appendix.

math.RA↗

arXiv2023

Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time

A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $\lambda$ tends to zero. In the literature, this problem has been addressed under various controllability or ergodicity conditions ensuring that the rescaled value function converges uniformly to a constant limit. In this case the limit can be characterized as the unique constant such that a suitable Hamilton-Jacobi equation has at least one continuous viscosity solution. In this paper, we study this problem without such conditions, so that the aforementioned limit needs not be constant. Our main result characterizes the uniform limit (when it exists) as the maximal subsolution of a system of Hamilton-Jacobi equations. Moreover, when such a subsolution is a viscosity solution, we obtain the convergence of optimal values as well as a rate of convergence. This mirrors the analysis of the discrete time case, where we characterize the uniform limit as the supremum over a set of sub-invariant half-lines of the dynamic programming operator. The emerging structure in both discrete and continuous time models shows that the supremum over sub-invariato half-lines with respect to the Lax-Oleinik semigroup/dynamic programming operator, captures the behavior of the limit cost as discount vanishes.

math.OC↗

arXiv2023

Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

The symmetrized tropical semiring is an extension of the tropical semifield, initially introduced to solve tropical linear systems using Cramer's rule. It is equivalent to the real tropical hyperfield, which has been used in the study of tropicalizations of semialgebraic sets. Polynomials over the symmetrized tropical semiring, and their factorizations, were considered by Quadrat. Recently, Baker and Lorscheid introduced a notion of multiplicity for the roots of univariate polynomials over hyperfields. In the special case of the hyperfield of signs, they related multiplicities with Descarte's rule of sign for real polynomials. We investigate here the factorizations of univariate polynomial functions over symmetrized tropical semirings, and relate them with the multiplicities of roots over these semirings. We deduce a Descartes' rule for "signs and valuations", which applies to polynomials over a real closed field with a convex valuation and an arbitrary (divisible) value group. We show in particular that the inequality of the Descartes' rule is tight when the value group is non-trivial. This extends to arbitrary value groups a characterization of Gunn in the rank one case, answering also to the tightness question. Our results are obtained using the framework of semiring systems introduced by Rowen, together with model theory of valued fields.

math.RA↗

arXiv2022

Semiring systems arising from hyperrings

Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. Conversely, we show that the systems arising in this way, called {\it hypersystems}, are characterized by certain elimination axioms. Systems are preserved under standard algebraic constructions; for instance matrices and polynomials over hypersystems are systems, but not hypersystems. We illustrate these results by discussing several examples of systems and hyperfields, and constructions like matroids over systems.

math.RA↗

arXiv2022

Computing Transience Bounds of Emergency Call Centers: a Hierarchical Timed Petri Net Approach

A fundamental issue in the analysis of emergency call centers is to estimate the time needed to return to a congestion-free regime after an unusual event with a massive arrival of calls. Call centers can generally be represented by timed Petri nets with a hierarchical structure, in which several layers describe the successive steps of treatments of calls. We study a continuous approximation of the Petri net dynamics (with infinitesimal tokens). Then, we show that a counter function, measuring the deviation to the stationary regime, coincides with the value function of a semi-Markov decision problem. Then, we establish a finite time convergence result, exploiting the hierarchical structure of the Petri net. We obtain an explicit bound for the transience time, as a function of the initial marking and sojourn times. This is based on methods from the theory of stochastic shortest paths and non-linear Perron--Frobenius theory. We illustrate the bound on a case study of a medical emergency call center.

math.OC↗

arXiv2021

Ambitropical geometry, hyperconvexity and zero-sum games

Shapley operators of undiscounted zero-sum two-player games are order-preserving maps that commute with the addition of a constant. We characterize the fixed point sets of Shapley operators, in finite dimension (i.e., for games with a finite state space). Some of these characterizations are of a lattice theoretical nature, whereas some other rely on metric or tropical geometry. More precisely, we show that fixed point sets of Shapley operators are special instances of hyperconvex spaces: they are sup-norm non-expansive retracts of $\R^n$, and also lattices in the induced partial order. Moreover, they retain properties of convex sets, with a notion of ``convex hull'' defined only up to isomorphism. This provides an effective construction of the injective hull or tight span, in the case of additive cones. For deterministic games with finite action spaces, these fixed point sets are supports of polyhedral complexes, with a cell decomposition attached to stationary strategies of the players, in which each cell is an alcoved polyhedron of $A_n$ type. We finally provide an explicit local representation of the latter fixed point sets, as polyhedral fans canonically associated to lattices included in the Boolean hypercube.

math.OC↗

arXiv2019

Tropical planar networks

We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary Jacobi matrices.

math.AC↗

arXiv2019

A Universal Approximation Result for Difference of log-sum-exp Neural Networks

We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtraction-free ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of Difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for design that possess a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing effective optimization-based design. We illustrate the proposed approach by applying it to data-driven design of a diet for a patient with type-2 diabetes.

cs.NE↗

arXiv2019

Matrix versions of the Hellinger distance

On the space of positive definite matrices we consider distance functions of the form $d(A,B)=\left[\tr\mathcal{A}(A,B)-\tr\mathcal{G}(A,B)\right]^{1/2},$ where $\mathcal{A}(A,B)$ is the arithmetic mean and $\mathcal{G}(A,B)$ is one of the different versions of the geometric mean. When $\mathcal{G}(A,B)=A^{1/2}B^{1/2}$ this distance is $\|A^{1/2}-B^{1/2}\|_2,$ and when $\mathcal{G}(A,B)=(A^{1/2}BA^{1/2})^{1/2}$ it is the Bures-Wasserstein metric. We study two other cases: $\mathcal{G}(A,B)=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$ the Pusz-Woronowicz geometric mean, and $\mathcal{G}(A,B)=\exp\big(\frac{\log A+\log B}{2}\big),$ the log Euclidean mean. With these choices $d(A,B)$ is no longer a metric, but it turns out that $d^2(A,B)$ is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of $m$ positive definite matrices with respect to these distance measures.

math-ph↗

arXiv2018

Log-sum-exp neural networks and posynomial models for convex and log-log-convex data

We show in this paper that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is an universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named LSET. Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST, which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.

cs.NE↗

arXiv2018

A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices

We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques.

math.OC↗

arXiv2017

Demand Response in the Smart Grid: the Impact of Consumers Temporal Preferences

In Demand Response programs, price incentives might not be sufficient to modify residential consumers load profile. Here, we consider that each consumer has a preferred profile and a discomfort cost when deviating from it. Consumers can value this discomfort at a varying level that we take as a parameter. This work analyses Demand Response as a game theoretic environment. We study the equilibria of the game between consumers with preferences within two different dynamic pricing mechanisms, respectively the daily proportional mechanism introduced by Mohsenian-Rad et al, and an hourly proportional mechanism. We give new results about equilibria as functions of the preference level in the case of quadratic system costs and prove that, whatever the preference level, system costs are smaller with the hourly mechanism. We simulate the Demand Response environment using real consumption data from PecanStreet database. While the Price of Anarchy remains always close to one up to 0.1% with the hourly mechanism, it can be more than 10% bigger with the daily mechanism.

math.OC↗

arXiv2017

Approximating the Volume of Tropical Polytopes is Difficult

We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $\alpha=2^{\text{poly}(m,n)}$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank. For tropical polytopes described by inequalities we prove that counting the number of integer points and calculating the volume are $\#$P-hard.

cs.CC↗

arXiv2017

Tropical compound matrix identities

We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues of the corresponding matrices. A tropical Sylvester-Franke identity is provided as well.

math.AC↗