arXiv research

New divide with gleams method simplifies symmetric link representation.

The paper characterizes links in 3D from divides with cusps.

This paper reveals hidden hyperbolic structures in divide links.

We give a method for constructing a shadowed polyhedron from a divide. The 4-manifold reconstructed from a shadowed polyhedron admits the structure of a Lefschetz fibration if it satisfies a certain property, which we call the LF-property. We will show that the shadowed polyhedron constructed from a divide satisfies th…

Starting from a divide, i.e. a generic immersion of finitely many copies of the interval [0,1] in the disk, we construct a classical link in the 3-sphere. We prove that the link's complement fibers over the circle, if the divide is connected. Moreover, we compute the monodromy diffeomorphism from the combinatorics of t…

New method to encode Weinstein 4-manifolds using multisections with divides.

Construct divide knots with specific genus properties.

I present a formula for the Casson invariant of knots associated with divides. The formula is written in terms of Arnold's invariants of pieces of the divide. Various corollaries are discussed.

Divides help construct fibered links from singularities.

Divide-and-conquer method splits large data sets for efficient analysis.

In the present paper we determine the Thurston-Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A.I. Stipsicz, we prove that the 3-manifold obtained from the 3-sphere by Dehn surger…

The study describes handle decompositions and Kirby diagrams for line arrangements.

This paper analyzes divide-and-conquer estimators for functional linear regression without assuming target function in the RKHS.

We give a complete characterization of a circle immersion that can be divided into two arc embeddings in terms of its chord diagram.

Divide knots and links, defined by A'Campo in the singularity theory of complex curves, is a method to present knots or links by real plane curves. The present paper is a continuation of the author's previous result that every knot in the major subfamilies of Berge's lens space surgery (i.e., knots yielding a lens spac…

A divide on an orientable 2-orbifold gives rise to a fibration of the unit tangent bundle to the orbifold.We characterize the corresponding monodromies as exactly the products of a left-veering horizontal and a right-veering vertical antitwist with respect to a cylinder decomposition, where the notion of an antitwist i…

Dividing deep learning models for consistent anomaly detection in changing log data.

A divide is a relative generic immersion of a finite union of copies of the unit interval in the unit disk. A divide defines a classical link in the 3- sphere, which is a fibered link if the image of the immersion is connected. We prove in this paper, that the Lefschetz number of the monodromy is 0. This result was kno…

Open books constructed from Morse functions and divides are shown to be isotopic.

In this article, we advance divide-and-conquer strategies for solving the community detection problem in networks. We propose two algorithms which perform clustering on a number of small subgraphs and finally patches the results into a single clustering. The main advantage of these algorithms is that they bring down si…

We propose a novel class of Sequential Monte Carlo (SMC) algorithms, appropriate for inference in probabilistic graphical models. This class of algorithms adopts a divide-and-conquer approach based upon an auxiliary tree-structured decomposition of the model of interest, turning the overall inferential task into a coll…

Dividing local Gaussian processes improve real-time prediction efficiency.

The paper is partially withdrawn: in its current form, Lemma 2.3 is false, so that our proof of Theorem A and Proposition B has an important gap. We were unable to fix it yet. Any help is most welcome. We prove that the restriction of surface minority to fiber surfaces of divides is a well-quasi-order. Here surface min…

In Divide & Recombine (D&R), big data are divided into subsets, each analytic method is applied to subsets, and the outputs are recombined. This enables deep analysis and practical computational performance. An innovate D\&R procedure is proposed to compute likelihood functions of data-model (DM) parameters for big dat…

We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatl…

The paper classifies all tight contact structures on a solid torus.

The primary objects of study in the ``knot theory of complex plane curves'' are C-links: links (or knots) cut out of a 3-sphere in the complex plane by complex plane transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links at…

DiCoLa recursively decomposes causal structure learning for latent variables.

Divide-and-conquer framework speeds up black-box inference for large data.

SwISS improves scalability of Bayesian inference for large datasets.

Tuning parameter selection is of critical importance for kernel ridge regression. To this date, data driven tuning method for divide-and-conquer kernel ridge regression (d-KRR) has been lacking in the literature, which limits the applicability of d-KRR for large data sets. In this paper, by modifying the Generalized Cr…

Divide-and-conquer is a general strategy to deal with large scale problems. It is typically applied to generate ensemble instances, which potentially limits the problem size it can handle. Additionally, the data are often divided by random sampling which may be suboptimal. To address these concerns, we propose the $DC^…

Divides state space into regions with identical term structure shapes.

We prove that Dranishnikov's $k$-dimensional resolution $d_k\colon μ^k\to Q$ is a UV$^{n-1}$-divider of Chigogidze's $k$-dimensional resolution $c_k$. This fact implies that $d_k^{-1}$ preserves $Z$-sets. A further development of the concept of UV$^{n-1}$-dividers permits us to find sufficient conditions for $d_k^{-1}(…

Study risk bounds for distributed ERM with general loss functions and hypothesis spaces.

Active learning (AL) repeatedly trains the classifier with the minimum labeling budget to improve the current classification model. The training process is usually supervised by an uncertainty evaluation strategy. However, the uncertainty evaluation always suffers from performance degeneration when the initial labeled …

Paper connects two portfolio methods, HRP and Minimum Variance, revealing their underlying similarity.

We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen seminorms of these curves. Using this calculation, we describe the fundamental polygon a…

We improve kernel ridge regression for skewed responses using oversampling and adaptive partitioning.

DS-Sync improves distributed DNN training efficiency by 94% with minimal accuracy loss.

We present a parallelized bijective graph matching algorithm that leverages seeds and is designed to match very large graphs. Our algorithm combines spectral graph embedding with existing state-of-the-art seeded graph matching procedures. We justify our approach by proving that modestly correlated, large stochastic blo…

We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the…

It is proved that every knot in the major subfamilies of J. Berge's lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a "divide knot" defined by N. A'Campo in the context of singularity theory of complex curves. For each knot given by Berge's parame…

A new distributed algorithm for fitting sparse additive models with feature division and decorrelation.

New method for efficient inference in large datasets.

We study the risk performance of distributed learning for the regularization empirical risk minimization with fast convergence rate, substantially improving the error analysis of the existing divide-and-conquer based distributed learning. An interesting theoretical finding is that the larger the diversity of each local…

New algorithm ranks players from partial comparisons with optimal rate.

Twinning splits data into fast, statistically similar sets.