We present extremal constructions connected with the property of simplicial collapsibility. (1) For each , there are collapsible (and shellable) simplicial -complexes with only one free face. Also, there are non-evasive -complexes with only two free faces. (Both results are optimal in all dimensions.) (2…
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Optimal transport is #P-hard when components are independent, even with approximate solutions.
problem Computational complexity of optimal transport with independent marginals.
method Proved #P-hardness and developed a pseudo-polynomial time approximation algorithm.
result Optimal transport is #P-hard even with independent components and approximate solutions.
In this study, we propose a new definition of multivariate conditional value-at-risk (MCVaR) as a set of vectors for discrete probability spaces. We explore the properties of the vector-valued MCVaR (VMCVaR) and show the advantages of VMCVaR over the existing definitions given for continuous random variables when adapt…
Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulasstat.ME
The study models insurance dependence using Bernstein copulas.
problem Modeling dependence structures in nonlife insurance data.
method Review and suggest fitting Bernstein copulas to empirical data.
result Monte Carlo simulation and PML estimation for aggregate losses.
In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differen…
A new model trains prior and encoder/decoder networks simultaneously for efficient generation.
problem Complex autoregressive prior in VQ-VAE models leads to slow generation.
method Builds a diffusion bridge between continuous and non-informative prior distributions.
result Model is competitive and efficient in optimization and sampling.
New tree-structured Markov fields with Poisson marginals for counting variables.
problem Counting variables with complex dependencies.
method Tree-structured Markov random fields with Poisson marginals.
result Straightforward sampling and joint probability calculations.
We propose a method for inferring the conditional indepen- dence graph (CIG) of a high-dimensional discrete-time Gaus- sian vector random process from finite-length observations. Our approach does not rely on a parametric model (such as, e.g., an autoregressive model) for the vector random process; rather, it only assu…
High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical me…
Paper studies gradient fields from discrete Morse functions for watershed-cut computation.
problem Computing watershed-cuts from discrete Morse functions.
method Discrete Morse Theory and simplicial stacks.
result Minimum Spanning Forest of dual graph is induced by gradient vector field.
Structures of optimal discrete gradient vector fields on surface with one or two critical cellsmath.DS
The study identifies all possible vector field structures on specific 2D shapes.
problem Optimal discrete gradient vector fields on surfaces with 1-2 critical cells.
method Analysis of discrete vector fields on 2D shapes with minimal critical cells.
result All possible structures of discrete Morse functions on specified shapes.
Develops combinatorial theory of vector bundles on simplicial complexes.
problem Creating a discrete theory for vector bundles and connections on simplicial complexes.
method Introduces discrete exterior covariant derivative and applies it to various geometric objects.
result Flat discrete connections yield a cochain complex computing twisted de Rham cohomology.
Two EM algorithms estimate prior distributions in mixture of linear regressions.
problem Estimating prior distributions in mixture of linear regressions.
method Two EM algorithms: one for continuous priors, one for discrete priors.
result Both algorithms accurately estimate prior distributions and the number of clusters.
Observations depending on sums of random variables are common throughout many fields; however, no efficient solution is currently known for performing max-product inference on these sums of general discrete distributions (max-product inference can be used to obtain maximum a posteriori estimates). The limiting step to …
After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, an…
The likelihood model of high dimensional data can often be expressed as , where is a collection of hidden features shared across objects, indexed by , and is a non-negative factor loading vector with entries where indicates the strength of …
DeformRS certifies deep networks against various input deformations.
problem Vulnerability of deep networks to input deformations.
method Randomized smoothing reformulation for general deformations.
result Certifies rich deformations including translations, rotations, scaling, and affine.
The paper analyzes the randomized midpoint method for Langevin diffusions, revealing biases and asymptotic properties.
problem Analyzing biases and asymptotic properties of the randomized midpoint method for Langevin diffusions.
method Characterization of stationary distribution and asymptotic normality for numerical integration.
result The step-size needs to go to zero for the method to be asymptotically unbiased.
Error estimates for discrete approximations of game options with multivariate diffusion asset pricesmath.PR
Study provides error estimates for approximating game options with diffusion asset prices.
problem Approximating fair prices of game options with diffusion asset prices.
method Error estimates for discrete approximations of diffusion processes, applied to game options.
result Effective tool for computing fair prices of game options in multi-asset markets.
Study optimal hedging for claims with random weights in discrete time.
problem Optimal hedging for claims with random weights in discrete time.
method Explicit recursive representation of optimal hedging strategy, without ND condition.
result Obtained explicit optimal hedging strategy in a recursive form.
A new method learns discrete representations for images and videos, improving upon previous models.
problem Learning discrete representations for images and videos to improve performance.
method Depthwise application of Vector Quantized Variational Autoencoders (VQVAE) to feature axis.
result 33% improvement in performance compared to previous discrete models.
This work develops discrete Gaussian models for vector-valued data on triangular meshes.
problem Discrete representation of continuous vector-valued environmental data.
method Develops discrete intrinsic Gaussian processes for vector-valued data on triangular meshes using discrete differential operators.
result Models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data.
Discrete Exterior Calculusmath.DG
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allow…
The reparameterization trick enables optimizing large scale stochastic computation graphs via gradient descent. The essence of the trick is to refactor each stochastic node into a differentiable function of its parameters and a random variable with fixed distribution. After refactoring, the gradients of the loss propag…
Exploratory Randomization for Discrete-Time Linear Exponential Quadratic Gaussian (LEQG) Problemmath.OC
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.
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…
Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.
problem Computing homology in discrete and smooth dynamical systems.
method Counting flow lines between orbits and critical points.
result Directly recovers homology from flow lines.
In this note, we derive concentration inequalities for random vectors with subGaussian norm (a generalization of both subGaussian random vectors and norm bounded random vectors), which are tight up to logarithmic factors.
Improved hierarchical discrete VAEs for better stability and performance.
problem Training stable and efficient hierarchical discrete VAEs with numerous latent variables.
method Introducing Relaxed-Responsibility Vector-Quantisation to parameterise discrete latent variables in a hierarchical structure.
result Achieved state-of-the-art bits-per-dim results for various standard datasets.
We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…
The paper studies how norms of random vectors are preserved by random projections.
problem Understanding how random matrix affects norms of random vectors.
method Proved the distribution of the norm of random vector is preserved by random projection.
result Random matrix preserves the distribution of the norm of random vectors with i.i.d. entries.
Multilayer switch networks are proposed as artificial generators of high-dimensional discrete data (e.g., binary vectors, categorical data, natural language, network log files, and discrete-valued time series). Unlike deconvolution networks which generate continuous-valued data and which consist of upsampling filters a…
Defines discrete differential geometry concepts in homotopy type theory.
problem No existing definition of Euler characteristic for comparison.
method Type families on higher inductive types, simplicial complexes, principal bundles, connections, curvature, vector fields, index.
result Theorem relating total curvature and total index, key to proving Gauss-Bonnet and Poincaré-Hopf theorems.
Physics-informed deep learning for PDEs solves forward and inverse problems efficiently.
problem Solving forward and inverse problems in parametric PDEs efficiently and accurately.
method Physics-informed deep latent variable model (PDDLVM) combining deep neural networks, probabilistic modelling, and variational inference.
result Achieves up to three orders of magnitude speed-up compared to traditional FEM while providing coherent uncertainty estimates.
In this paper, we face the problem of simulating discrete random variables with general and varying distributions in a scalable framework, where fully parallelizable operations should be preferred. The new paradigm is inspired by the context of discrete choice models. Compared to classical algorithms, we add paralleliz…
Given a triangulated region in the complex plane, a discrete vector field assigns a vector to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…
Paper introduces privacy-preserving few-shot learning for images.
problem Privacy risk in few-shot learning systems.
method Discrete embedding vectors and one-way hash functions.
result Achieves computational pan privacy without storing embeddings.
Study forecasts Bitcoin prices using ML algorithms.
problem Accurately predicting Bitcoin price movements.
method Applied four ML algorithms: SVM, ANN, NB, RF, and LR.
result RF outperforms other models in continuous dataset, NB in discrete.
VQ-DRAW compresses images and generates realistic samples.
problem Learning compact discrete representations of images.
method Sequential discrete VAE with vector quantization.
result VQ-DRAW effectively compresses and generates images.
1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the spa…
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
New estimator reduces variance in discrete random variables.
problem Estimating gradients for discrete random variables with reduced variance.
method Sampling without replacement and Rao-Blackwellization.
result Our estimator is the most consistent gradient estimator across different entropy settings.
We develop time-uniform confidence spheres for estimating means of random vectors.
problem Sequential mean estimation in high-dimensional spaces.
method Derive time-uniform confidence sphere sequences (CSSs) for various types of random vectors.
result Optimal CSSs for log-concave, sub-Gaussian, and sub- random vectors.
Simplified analysis of diffusion models using discrete random variables.
problem Theoretical analysis of diffusion models is complex and requires rigorous proofs.
method Simplified framework for analyzing Euler--Maruyama discretization of VP-SDEs using Grönwall's inequality.
result Standard Gaussian noise can be replaced by discrete random variables without sacrificing convergence guarantee.
Paper proposes a new estimator for generic discrete distributions.
problem Estimating gradients for stochastic nodes in deep generative models.
method Generalized Gumbel-Softmax estimator using truncation, Gumbel-Softmax trick, and linear transformation.
result Efficacy and practical value demonstrated in synthetic examples and topic models.
Study improves error bounds for sparse regression with heavy-tailed covariates.
problem Estimating sparse coefficients in linear regression with heavy-tailed covariates.
method Employed an -penalized Huber regression method.
result Error bound identical to Gaussian case for -subexponential covariates.
Random square-tiled surfaces have normal genus distribution and cover all integer vectors.
problem Distribution and properties of random square-tiled surfaces.
method Randomizing model and local central limit theorem for genus.
result The distribution of the genus is asymptotically normal and contains all primitive integer vectors.
Generative Assignment Flows for Representing and Learning Joint Distributions of Discrete Datastat.ML
Generative model for joint discrete distributions using randomized assignment flows.
problem Efficiently representing and sampling from complex joint distributions of discrete variables.
method Randomized assignment flows on the statistical submanifold of factorizing distributions.
result Our model can efficiently represent and sample from any target distribution and assess likelihood of unseen data points.