arXiv research

CMRFs extend PGMs for topological data, capturing both conditional and marginal dependencies.

Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.

Enhances topology optimization with multiclass microstructures using latent variable Gaussian process.

A novel method optimizes variable-stiffness structures for better strength and weight.

Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomi…

GeoPhy uses geometric gradients to efficiently infer phylogenetic trees from molecular data.

A new model encodes distances and topology in latent variables.

We give some characterizations for the critical values at infinity of a rational function in two complex variables in terms of the Euler characteristic, the Malgrange condition and the M-tameness

In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.

We present an empirical analysis of the network formed by the trade relationships between all world countries, or World Trade Web (WTW). Each (directed) link is weighted by the amount of wealth flowing between two countries, and each country is characterized by the value of its Gross Domestic Product (GDP). By analysin…

The following numerical control over the topological equivalence is proved: two complex polynomials in $n\not= 3$ variables and with isolated singularities are topologically equivalent if one deforms into the other by a continuous family of polynomial functions $f_s \colon \mathbb{C}^n \to \mathbb{C}$ with isolated sin…

The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds w…

This paper finds efficient algorithms for approximating Markov networks with k-tree topologies.

We prove that the space of complex irreducible polynomials of degree $d$ in $n$ variables satisfies two forms of homological stability: first, its cohomology stabilizes as $d$ increases, and second, its compactly supported cohomology stabilizes as $n$ increases. Our topological results are inspired by counting results …

New model uncovers non-Euclidean neural representations.

Topological entropy decreases strictly along Ricci flow near hyperbolic metrics.

A standard Variational Autoencoder, with a Euclidean latent space, is structurally incapable of capturing topological properties of certain datasets. To remove topological obstructions, we introduce Diffusion Variational Autoencoders with arbitrary manifolds as a latent space. A Diffusion Variational Autoencoder uses t…

We propose a topological learning algorithm for the estimation of the conditional dependency structure of large sets of random variables from sparse and noisy data. The algorithm, named Maximally Filtered Clique Forest (MFCF), produces a clique forest and an associated Markov Random Field (MRF) by generalising Prim's m…

We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system $x' = f(x,y,ε), y' = εg(x,y,ε)$ with one-dimensional slow variable $y$. Our validation procedure is based on topological tools called isolatin…

Topology design optimization offers tremendous opportunity in design and manufacturing freedoms by designing and producing a part from the ground-up without a meaningful initial design as required by conventional shape design optimization approaches. Ideally, with adequate problem statements, to formulate and solve the…

Introduces TSI, a variance-based measure for persistence barcodes.

PhyloVAE learns tree topologies without supervision.

Machine learning provides algorithms that can learn from data and make inferences or predictions on data. Bayesian networks are a class of graphical models that allow to represent a collection of random variables and their condititional dependencies by directed acyclic graphs. In this paper, an inference algorithm for …

Stable topological summary captures evolving dependency structure in dynamic Bayesian networks.

We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometr…

We provide a theoretical analysis of the representation learning problem aimed at learning the latent variables (design matrix) $Θ$ of observations $Y$ with the knowledge of the coefficient matrix $X$. The design matrix is learned under the assumption that the latent variables $Θ$ are smooth with respect to a (known) t…

A semi-parametric, non-linear regression model in the presence of latent variables is applied towards learning network graph structure. These latent variables can correspond to unmodeled phenomena or unmeasured agents in a complex system of interacting entities. This formulation jointly estimates non-linearities in the…

Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally defined by M. Farber.

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…

This paper introduces TDA and TSI for better business analytics.

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogu…

Among the proposed network models, the hidden variable (or good get richer) one is particularly interesting, even if an explicit empirical test of its hypotheses has not yet been performed on a real network. Here we provide the first empirical test of this mechanism on the world trade web, the network defined by the tr…

Enhanced Zika spread forecasting using topological data analysis.

While a wide range of interpretable generative procedures for graphs exist, matching observed graph topologies with such procedures and choices for its parameters remains an open problem. Devising generative models that closely reproduce real-world graphs requires domain knowledge and time-consuming simulation. While e…

Proposes a new method for disentangling data representations using topological analysis.

Globalizes Jones and Alexander polynomials using topological intersections.

In this paper, a class of statistics named ART (the alternant recursive topology statistics) is proposed to measure the properties of correlation between two variables. A wide range of bi-variable correlations both linear and nonlinear can be evaluated by ART efficiently and equitably even if nothing is known about the…

Categorical d-separation criterion simplifies probability graph analysis.

The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…

Traders adopt different trading strategies to maximize their returns in financial markets. These trading strategies not only results in specific topological structures in trading networks, which connect the traders with the pairwise buy-sell relationships, but also have potential impacts on market dynamics. Here, we pr…

This paper develops a theory of graded manifolds in differential geometry.

In recent years, twisted Alexander polynomial has been playing an important role in low-dimensional topology. For Montesinos links, we develop an efficient method to compute the twisted Alexander polynomial associated to any linear representation. In particular, formulas for multi-variable Alexander polynomials of thes…

This chapter covers methods for identifying and inferring graph topologies.

This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolutionary trees. The …

Constructs new topological theories in 2D not fitting standard axioms.

A new framework uses an Incremental Transformer to design geopolymer mixtures efficiently.

The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic proce…

Ensemble decoders to capture latent space topology in deep generative models.