Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
Discrete approximation solves Björling's minimal surface problem.
problem Constructing minimal surfaces from real-analytic curves with specified normal fields.
method Approximate solution by discrete minimal surfaces and discrete isothermic surfaces.
result Approximation error is proportional to the square of the mesh size.
In this paper, we develop several related finite dimensional variational principles for discrete optimal transport (DOT), Minkowski type problems for convex polytopes and discrete Monge-Ampere equation (DMAE). A link between the discrete optimal transport, discrete Monge-Ampere equation and the power diagram in computa…
Study tackles discretization problem in crafting adversarial examples for discrete integer domains.
problem Discretization problem in crafting adversarial examples for discrete integer domains.
method Proposes a black-box method to reduce adversarial example searching to a derivative-free optimization problem.
result Significantly higher success rate in crafting adversarial images in discrete integer domain compared to black-box methods.
Proves hardness of semi-discrete optimal transport and proposes regularization methods.
problem Computing Wasserstein distance between discrete and non-discrete probability measures.
method Proves hardness, introduces distributionally robust dual optimal transport, regularizes primal objective, uses stochastic gradient descent.
result Regularization schemes and improved convergence guarantees for semi-discrete optimal transport problems.
New discrete models for constant mean curvature surfaces and tori.
problem Creating discrete models for constant mean curvature surfaces and tori.
method Integrable theory of discrete polarised curves and Darboux transforms.
result Closed-form discrete parametrisations of discrete isothermic cylinders, discrete constant mean curvature cylinders, and discrete isothermic tori.
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propos…
Solves a discrete logarithmic Minkowski problem for electrostatic p-capacity.
problem Characterize measures generated by electrostatic p-capacity.
method Solves the discrete logarithmic Minkowski problem for 1 < p < n.
result Solves the discrete logarithmic Minkowski problem for measures in general position.
Drawing a sample from a discrete distribution is one of the building components for Monte Carlo methods. Like other sampling algorithms, discrete sampling suffers from the high computational burden in large-scale inference problems. We study the problem of sampling a discrete random variable with a high degree of depen…
Investment strategy optimization from discrete to continuous models.
problem Optimizing investment strategies and stopping times in both continuous and discrete settings.
method Characterized value functions via quadratic reflected BSDEs for continuous case, discretized BSDEs for discrete case, and derived uniform convergence rates.
result Uniform convergence and rate from discrete to continuous quadratic reflected BSDEs.
Proposes hybrid reinforcement learning for both discrete and continuous control problems.
problem Real-world control problems involving both discrete and continuous decision variables.
method Solves hybrid problems by optimizing for discrete and continuous actions simultaneously.
result Efficiently solves hybrid reinforcement learning problems and improves upon expert heuristics.
The paper tackles fast rates in structured prediction problems.
problem Structured prediction problems with discrete outputs.
method Introducing continuous surrogate problems and leveraging their convergence rates for discrete problems.
result Super fast rates, including exponential rates, for excess risk in structured prediction problems.
RL solves discrete LQ control with Gaussian optimal policy.
problem Discrete-time linear-quadratic control problem.
method Entropy-based RL to find Gaussian optimal policy.
result RL algorithm solves mean-variance asset-liability management problem.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
We study robust stochastic optimization problems in the quasi-sure setting in discrete-time. The strategies in the multi-period-case are restricted to those taking values in a discrete set. The optimization problems under consideration are not concave. We provide conditions under which a maximizer exists. The class of …
Solves risk-aware optimal switching problems in discrete time.
problem Non-Markovian optimal switching problems with risk awareness and general filtration.
method Solves reflected backward stochastic difference equations.
result Existence and uniqueness of solutions for the problems.
New method approximates Gaussian curvature on discrete surfaces.
problem Approximating solutions to the prescribed Gaussian curvature problem.
method Discrete conformality and convex functional minimization.
result Efficient numerical method to compute solutions.
We present a new approach for studying the problem of optimal hedging of a European option in a finite and complete discrete-time market model. We consider partial hedging strategies that maximize the success probability or minimize the expected shortfall under a cost constraint and show that these problems can be trea…
This paper tackles convex-submodular minimax problems in mixed continuous-discrete domains.
problem Convex-submodular minimax problems in mixed continuous-discrete domains.
method Introduces new notions of optimality and proposes iterative algorithms combining discrete and continuous optimization.
result Characterizes convergence rates, computational complexity, and quality of solutions for convex and monotone-submodular minimax problems.
Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invar…
This paper studies the properties of discrete time stochastic optimal control problems associated with portfolio selection. We investigate if optimal continuous time strategies can be used effectively for a discrete time market after a straightforward discretization. We found that Merton's strategy approximates the per…
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
A method for disentangling discrete and continuous factors of data without using a discriminator network.
problem Unsupervised disentanglement of discrete and continuous factors of data.
method A procedure that minimizes total correlation of continuous latent variables and a separate discrete inference procedure.
result The method significantly outperforms current disentanglement methods based on disentanglement score and inference network classification score.
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
Proposes DAM for optimizing discrete generative models.
problem Challenges in optimizing discrete generative models.
method Discrete Adjoint Matching (DAM) for discrete state spaces.
result Demonstrates effectiveness on synthetic and mathematical reasoning tasks.
Paper tackles decidability of subgroup discreteness problem.
problem Decidability of finitely generated subgroup discreteness in PSL(2,R) and PSL(2,C). method Examines different computational models to determine if the discreteness problem is decidable.
result The answer depends on the model of computation chosen.
We explore a new method for discrete-time control problems using randomization and entropy.
problem Discrete-time linear-exponential quadratic Gaussian (LEQG) control problem.
method Introduce exploration through randomization and apply duality between free energy and relative entropy.
result Reduced LEQG problem to equivalent risk-neutral LQG control problem with entropy regularization.
New formulae identify discrete probability laws without needing normalization constants.
problem Characterizing non-normalized discrete probability distributions.
method Derive explicit formulae for mass functions using Stein's method.
result Developed tools for solving statistical problems without normalization constants.
In this article, we study an analog of the Björling problem for isothermic surfaces (that are more general than minimal surfaces): given a real analytic curve γ in R3, and two analytic non-vanishing orthogonal vector fields v and w along γ, find an isothermic surface that is tangent to γ and that…
Efficient deep policy gradient method for continuous-time control problems.
problem Optimal control in continuous time with fine time discretization.
method Multi-scale deep policy gradient method with varying time discretization.
result Targeted efficiency in computational resources achieved through multi-scale approach.
Locally-verifiable conditions ensure exactness of spline discrete de Rham complex.
problem Ensuring cohomological equivalence of spline discrete complex to continuous de Rham complex.
method Theoretical analysis and locally-verifiable sufficient conditions for exactness.
result Locally-verifiable conditions guarantee exactness of hierarchical B-spline discrete de Rham complex.
Bayesian optimization tackles mixed discrete-continuous problems with Gaussian processes.
problem Optimizing problems with both discrete and continuous variables using costly simulations.
method Relaxing discrete variables into continuous latent variables, using Bayesian optimization, and incorporating compatibility constraints with Lagrangians.
result Comparative analysis of different mixed Bayesian optimization approaches.
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
The study examines discrete states in hyperbolic spaces using specific transformations.
problem Characterizing discrete states in hyperbolic spaces via specific transformations.
method Analyzing two-parameter families of subgroups in hyperbolic planes and spaces with up to four generators.
result Discreteness of accessible states in hyperbolic spaces is determined for specific transformations.
Study scaling limits of utility indifference prices in discretized Bachelier model.
problem Analyzing utility indifference prices for path-dependent European options in a discretized Bachelier model.
method Purely probabilistic approach, including duality argument, optimal drift control problem, martingale techniques, and strong invariance principles.
result Obtained a scaling limit for utility indifference prices as the number of trading times increases.
We establish a connection between recent developments in the study of vortices in the abelian Higgs models, and in the theory of structure-preserving discrete conformal maps. We explain how both are related via conformal mapping problems involving prescribed linear combinations of the curvature and volume form, and sho…
In this article we discuss the problem of calculating optimal model-independent (robust) bounds for the price of Asian options with discrete and continuous averaging. We will give geometric characterisations of the maximising and the minimising pricing model for certain types of Asian options in discrete and continuous…
Discrete diffusion samplers improve sampling from unnormalised densities.
problem Sampling from discrete unnormalised densities efficiently.
method Introduce off-policy training techniques and data-to-energy Schrödinger bridge training for discrete diffusion samplers.
result Improved performance on synthetic and new benchmarks.
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.
problem Discovering discrete algorithmic axioms missing in deep learning.
method Cayley-table completion as a testbed for algorithmic complexity minimization.
result Formal exact recovery bounds for Cayley-table completion.
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
Adaptive discretization improves model-based RL in large spaces.
problem Efficient model-based reinforcement learning in large state-action spaces.
method Optimistic one-step value iteration with adaptive discretization.
result Adaptive discretization leads to better performance and lower memory usage.
Exact Bayesian inference for discrete models using probability generating functions.
problem Discrete statistical models with infinite support and continuous priors.
method Probabilistic programming language with automatic differentiation and probability generating functions.
result Genfer tool provides exact solutions for a wide range of inference problems.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
DNFS trains efficient samplers for discrete distributions using locally equivariant Transformers.
problem Sampling from unnormalised discrete distributions.
method DNFS learns a rate matrix to satisfy the Kolmogorov equation, using control variates and locally equivariant Transformers.
result DNFS achieves efficient and effective sampling across various applications.
The paper proposes a variational autoencoder for discrete data analysis.
problem Sparse, high-dimensional, and overdispersed discrete data analysis.
method Variational autoencoder based on negative-binomial distribution.
result The proposed models achieve significantly better performance on text analysis, collaborative filtering, and multi-label learning compared to state-of-the-art baselines.
Deep learning techniques have been hugely successful for traditional supervised and unsupervised machine learning problems. In large part, these techniques solve continuous optimization problems. Recently however, discrete generative deep learning models have been successfully used to efficiently search high-dimensiona…