arXiv research

Unified analysis for decentralized SGD across various topologies and updates.

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…

This article was originally published in Topology 31 (1992). The present hyperTeXed redaction corrects a few typographical errors and updates the references.

Enhances GNNs by improving input data quality from topology and labels.

A new method reduces communication costs in decentralized optimization.

New research shows sparse topologies can lead to faster convergence in distributed optimization.

We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topolog…

This is an almost self-contained monograph (containing some new results) on left-orderable groups which mostly rely on dynamical and probabilistic aspects, but also on geometric, combinatorial, analytic, and topological ones. This new version contains many improvements, corrections and updates, many of them suggested b…

Study of profinite quandles with constructions and characterizations.

This note presents the handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, yielding a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented 4-manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topo…

This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…

This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective …

SOM-VQ tokenizes discrete models with semantic structure and navigable topology.

We give an updated extended survey of results related to the celebrated unsolved generalized R. L. Moore problem. In particular, we address the problem of characterizing codimension one manifold factors, i.e. spaces $X$ having the property that $X \times \mathbb{R}$ is a topological manifold. A main part of the paper i…

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how traj…

AdaCGP learns dynamic graph topology from time series data, improving over existing methods.

We present a distributed (non-Bayesian) learning algorithm for the problem of parameter estimation with Gaussian noise. The algorithm is expressed as explicit updates on the parameters of the Gaussian beliefs (i.e. means and precision). We show a convergence rate of $O(1/k)$ with the constant term depending on the numb…

A new method for optimal filtration learning in time-series data analysis.

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

This survey of some of the more topological aspects of the placement problem for complex curves in complex surfaces was originally published in L'Enseignement Mathematique 29 (1983). The present LaTeXed redaction corrects several typographical errors without, I hope, introducing new ones; some minor emendations and ref…

This paper optimizes portfolios using TDA and financial news sentiment.

Over 50 years of work on group actions on $4$-manifolds, from the 1960's to the present, from knotted fixed point sets to Seiberg-Witten invariants, is surveyed. Locally linear actions are emphasized, but differentiable and purely topological actions are also discussed. The presentation is organized around some of the …

CAGNN learns graph embeddings without labels by clustering and refining graph topology.

Improves decentralized learning by teleporting active nodes for better convergence.

Enhances graph neural networks with structural message-passing for better generalization.

Unified framework recovers exact input from SOM activation patterns.

Many applications require sparse neural networks due to space or inference time restrictions. There is a large body of work on training dense networks to yield sparse networks for inference, but this limits the size of the largest trainable sparse model to that of the largest trainable dense model. In this paper we int…

This paper considers the problem of adaptively searching for an unknown target using multiple agents connected through a time-varying network topology. Agents are equipped with sensors capable of fast information processing, and we propose a decentralized collaborative algorithm for controlling their search given noisy…

This paper updates knot invariants using Hopf algebras and categorifies their structure.

Optimizes structure topology for ductile and brittle fracture resistance.

Paper proposes efficient weight updates for edge nodes with minimal communication.

A new update rule for deep reinforcement learning reduces learning variance and variance in reference signals.

This paper proposes the Mesh Neural Network (MNN), a novel architecture which allows neurons to be connected in any topology, to efficiently route information. In MNNs, information is propagated between neurons throughout a state transition function. State and error gradients are then directly computed from state updat…

New algorithm constrains SOMs to create supervised low-dimensional mappings.

Paper proves Jeffrey's update rule minimizes relative entropy.

AMUSE uses reinforcement learning to predict optimal model updates.

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…

We shed new insights on the two commonly used updates for the online $k$-PCA problem, namely, Krasulina's and Oja's updates. We show that Krasulina's update corresponds to a projected gradient descent step on the Stiefel manifold of the orthonormal $k$-frames, while Oja's update amounts to a gradient descent step using…

Efficiently updates posterior tree distributions over meta-trees.

Recently, the technique of local updates is a powerful tool in centralized settings to improve communication efficiency via periodical communication. For decentralized settings, it is still unclear how to efficiently combine local updates and decentralized communication. In this work, we propose an algorithm named as L…

The paper examines how updates to probabilistic models influence behavior based on evidence.

Paper improves policy updates in reinforcement learning to speed up learning.

In this paper, we study the randomized distributed coordinate descent algorithm with quantized updates. In the literature, the iteration complexity of the randomized distributed coordinate descent algorithm has been characterized under the assumption that machines can exchange updates with an infinite precision. We con…

Proposes a new method for nonlinear Bayesian updates using ensemble kernel regression.

This paper analyzes how periodic and soft target updates stabilize linear Q-learning.

In this letter, we generalize the convolutional NMF by taking the $β$-divergence as the contrast function and present the correct multiplicative updates for its factors in closed form. The new updates unify the $β$-NMF and the convolutional NMF. We state why almost all of the existing updates are inexact and approximat…

Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties. We give an efficient method to compute those $q$-series that come from planar gra…

EnKF's update is shown to be similar to Matheron's method in Gaussian process regression.