Study examines persistence diagrams in machine learning, proposing permutation tests.
problem Understanding the power and limitations of persistence diagrams in machine learning.
method Carried out experiments on graph and shape data, proposed permutation tests for persistence diagrams.
result Persistence pairing shows significant improvement in various tasks, but the most critical values are most discriminative.
Deep networks partition input space into regions with complex affine mappings.
problem Understanding the geometry and complexity of deep neural networks.
method Analyzed deep networks as max-affine spline operators and power diagrams.
result Composition of deep networks results in a progressively subdivided power diagram.
This paper classifies curves in genus two handlebodies.
problem Classifying curves in the boundary of a genus two handlebody.
method Using R-R diagrams and cutting disks, the paper classifies curves as primitive, proper power, or Seifert.
result The paper provides all possible R-R diagrams for these curves.
This paper classifies knots with simple curves in genus 2 handlebodies.
problem Classifying knots with specific curves in handlebodies.
method Analyzing R-R diagrams and 2-handle additions.
result Classification of knots with proper power curves in genus 2 handlebodies.
STRAND: A single representation for hypothesis testing and vectorisation of persistence diagrams
problem Comparing persistence diagrams
method Survival topological representation analysis
result Non-parametric two-sample test with calibrated Type I error and high power
We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of q-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of …
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …
Revises SWK for persistence diagrams using Figalli-Gigli distance.
problem Efficiently embedding persistence diagrams in a Hilbert space.
method Directly use Figalli-Gigli distance to build a positive definite kernel.
result SFGK shares properties with SWK and performs similarly on benchmarks.
Many applications in data analysis begin with a set of points in a Euclidean space that is partitioned into clusters. Common tasks then are to devise a classifier deciding which of the clusters a new point is associated to, finding outliers with respect to the clusters, or identifying the type of clustering used for th…
ENN neural network learns logical syllogisms using Euler diagrams.
problem Traditional neural networks struggle with logical reasoning, especially syllogisms.
method ENN represents logical relations as Euler diagrams, optimizing syllogism structures with a novel back-propagation algorithm.
result ENN can precisely represent and reason with all 24 syllogism structures.
Improved modeling of persistence diagrams for data analysis.
problem Determining significant outliers in persistence diagrams.
method Modification of the RST (Replicating Statistical Topology) model using MCMC Metropolis-Hastings algorithm.
result The modified RST model improves the goodness of fit in persistence diagram analysis.
Unified model for knot polynomials using quantum Heegaard diagrams.
problem Categorify knot polynomials using Floer homology.
method Construct quantum Heegaard diagrams, identify gradings, and define a two-variable graded intersection.
result Unified intersection model recovers Alexander and Jones polynomials.
New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
MXGNet tackles visual reasoning tasks using graph neural networks.
problem Abstract reasoning, especially in the visual domain, is challenging for AI.
method Combines object-level representations, graph neural networks, and multiplex graphs.
result Achieves state-of-the-art accuracy on Euler Diagram Syllogisms and outperforms state-of-the-art models on RPM datasets.
Introduces info intervention to handle causal questions and check counterfactual variables.
problem Controversial interpretation of causal questions for non-manipulable variables and lack of power to check counterfactual variables.
method Intervenes input/output information of causal mechanisms, providing causal diagrams for communication and theoretical focus.
result Causal diagrams based on info intervention provide a new perspective on information transfer as causality.
Study of spaces of pure braids and string links using diagrams and integrals.
problem Understanding spaces of pure braids and string links through algebraic structures.
method Use of Kontsevich's CDGA of diagrams and Chen's iterated integrals to establish Hopf algebra isomorphisms and connections.
result Established a correspondence between Milnor invariants and Chen integrals for Brunnian spherical links.
Polynomials derived from Heegaard diagrams for 3-manifolds.
problem Creating polynomial invariants for 3-manifolds.
method Using Heegaard diagrams and cellularly embedded graphs to define polynomials.
result Finiteness of minimal Heegaard graphs for a fixed manifold enables polynomial definition.
A new method uses vectorized summaries of persistence diagrams for efficient hypothesis testing.
problem Efficient hypothesis testing for large and complex persistence diagrams.
method Vectorized summaries of Betti functions and a new shuffling technique.
result The vectorized Betti function leads to competitive results compared to baseline methods.
Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare. This is currently the best possible analogue of a pair of theorems on alternati…
Investors in stock market are usually greedy during bull markets and scared during bear markets. The greed or fear spreads across investors quickly. This is known as the herding effect, and often leads to a fast movement of stock prices. During such market regimes, stock prices change at a super-exponential rate and ar…
For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations V(t), we decompose the tensor powers of V(t) into irreducible factors in a uniform manner, using a tool we call {\it diagram induction}. In particular, we interpret the decompostion formulas of Deligne \cite{d…
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…
The paper connects quantum invariants to intersections of Lagrangians in symmetric power spaces.
problem Computing colored Jones and Alexander polynomials.
method Using two Lagrangians in a symmetric power of a surface to compute polynomials.
result Colored Jones and Alexander polynomials are special cases of a graded intersection between Lagrangians.
Develops a new framework to analyze gradient flow regimes and derive explicit solutions.
problem Analyzing scaling regimes and deriving explicit analytic solutions for gradient flow in large learning problems.
method Formal power series expansion of the loss evolution with coefficients encoded by diagrams.
result Reveals different learning phases and obtains explicit solutions in some cases.
The author recently proved the existence of an infinite order cork: a compact, contractible submanifold C of a 4-manifold and an infinite order diffeomorphism f of ∂C such that cutting out C and regluing it by distinct powers of f yields pairwise nondiffeomorphic manifolds. The present paper exhibits …
Two graph homologies help compute embedding space.
problem Computing the rational homotopy group of long embeddings.
method Invented two graph homologies and constructed a map between them.
result A monomorphism from top hairy graph homology to top BCR graph homology.
Characterizes local tropicalizations of splice type surface singularities.
problem Understanding splice type surface singularities from a tropical geometry perspective.
method Characterization of local tropicalizations as cones over splice diagrams, using tropical methods.
result Characterizes local tropicalizations of splice type surface singularities as cones over associated splice diagrams.
New algorithm speeds up group equivariant neural networks computations.
problem Challenging computations in group equivariant neural networks.
method Diagrammatic framework based on category theory for matrix multiplication.
result Exponential improvement in time complexity for matrix multiplication.
We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recove…
The paper refines transformations of lattice diagrams and introduces dotted diagrams.
problem Investigating transformations and deformations of lattice diagrams and their associated dotted diagrams.
method Introducing dotted diagrams and investigating deformations of these diagrams, relating them to transformations of lattice diagrams.
result Refined results on the relation between deformations of admissible dotted diagrams and transformations of lattice diagrams.
This note explains how to transform Heegaard diagrams into framed link diagrams.
problem No specific problem stated; transformation of diagrams is the focus.
method Explains a procedure to transform Heegaard diagrams into framed link diagrams.
result Demonstrates a method to transform Heegaard diagrams into framed link diagrams.
New minimal link diagrams found, including torus links and homogeneous ones.
problem Finding minimal link diagrams with new classes.
method Morton-Franks-Williams inequality approach.
result New classes of minimal link diagrams, including previously unproven ones.
We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…
The influence of Commodity Trading Advisors (CTA) on the price process is explored with the help of a simple model. CTA managers are taken to be Kelly optimisers, which invest a fixed proportion of their assets in the risky asset and the remainder in a riskless asset. This requires regular adjustment of the portfolio w…
New method estimates causal effects in complex spaces using topological structures.
problem Challenges in estimating causal effects in non-Euclidean spaces.
method Developed a topological causal inference framework using power-weighted silhouette functions of persistence diagrams.
result Successfully quantifies topological treatment effects across various complex outcomes.
Algorithm converts Kirby diagrams to trisection diagrams for 4-manifolds.
problem Creating efficient trisection diagrams for 4-manifolds.
method Algorithm converting Kirby diagrams to trisection diagrams.
result Provides examples of trisection diagrams for 4-manifolds.
Kernelized Taylor diagram visualizes data populations with fewer assumptions.
problem Limitations of Taylor diagram in capturing non-linear relationships and sensitivity to outliers.
method Proposes a kernelized version of the Taylor diagram that uses maximum mean discrepancy and kernel mean embedding.
result Kernelized Taylor diagram visualizes data populations with minimal assumptions of data distributions.
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative.In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram by use of the double covering …
Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.
We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent diagrams. We conjecture that these functions are rational functions of the exponentials …
A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links.In this paper, we introduce a method of converting a virtual link d…
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of …
Problems on region choices for knot and link diagrams solved using Alexander numbering.
problem Existence of solutions for region choice problems on knot and link diagrams.
method Alexander numbering for regions, alternative proofs, necessary and sufficient conditions.
result Existence of solutions for region choice problems on link diagrams.
Table of symmetric diagrams for knots up to 10 crossings.
problem Finding symmetric diagrams for strongly invertible knots.
method Compilation of symmetric diagrams for knots up to 10 crossings.
result Similarity of transversal diagrams to symmetric union diagrams for strongly invertible knots.
The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we provide a framework for working with subknots inside of knot diagrams via knotoid dia…
Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
Rectangular diagrams help analyze foliations in 3-sphere.
problem Analyzing foliations in 3-sphere complements.
method Introduced rectangular diagrams for foliations and links.
result Any co-orientable finite depth foliation can be presented by a compatible rectangular diagram.