Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
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48 results for “mean squared linking number”
In order to model entanglements of polymers in a confined region, we consider the linking numbers and writhes of cycles in random linear embeddings of complete graphs in a cube. Our main results are that for a random linear embedding of in a cube, the mean sum of squared linking numbers and the mean sum of square…
Study on linking numbers in random book embeddings of complete graphs.
problem Distribution and mean of linking numbers in random book embeddings of complete graphs.
method Analyzes a family of two-component links arising from random embeddings of complete graphs, using Eulerian numbers and linear growth in mean linking number.
result Mean of squared linking number over all random embeddings is $ rac{i}{6}$, where is the number of interior edges.
New homotopy types defined for links in thickened surfaces with higher genus.
problem Defining stable homotopy types for links in surfaces with higher genus.
method Defined Khovanov-Lipshitz-Sarkar homotopy types and Steenrod squares for links in thickened surfaces with genus > 1.
result First meaningful Khovanov-Lipshitz-Sarkar stable homotopy types for links in 3-manifolds other than the 3-sphere.
Paper provides statistical guarantees for GNNs in link prediction.
problem Link prediction accuracy in graph neural networks.
method Proposes a linear GNN architecture (LG-GNN) and derives statistical guarantees.
result LG-GNN produces consistent estimators for edge probabilities and has better detection of high-probability edges.
New method optimizes tail dependence coefficient estimation.
problem Estimating tail dependence in nonparametric data.
method Optimal threshold selection combining mean squared error and copula estimation.
result Improved accuracy in tail dependence coefficient estimation.
For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of t…
The paper generalizes linking number properties for complete graphs.
problem Understanding linking numbers in spatial complete graphs.
method Analyzing the sum of square linking numbers and triangle-triangle links.
result Explicit formula for the sum of square linking numbers in large complete graphs.
Study on typical knots and links using grid diagrams, focusing on size, components, and writhe.
problem Understanding the statistical behavior of knots and links, especially their typical properties.
method Modeling knots and links with grid diagrams, examining three invariants: size, components, and writhe, through numerical analysis.
result The size of a random knot is uniformly distributed and linearly dependent on grid size, while the number of components follows a distribution whose mean and variance grow with log_2 of grid size.
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
The paper provides bounds for the ropelength of a link in terms of the crossing numbers of its split components. As in earlier papers, the bounds grow with the square of the crossing number; however, the constant involved is a substantial improvement on previous results. The proof depends essentially on writing links i…
Study on LMMSE estimation with model mismatch, quantifying MSE trade-offs.
problem Model mismatch in LMMSE estimation with undermodeling.
method Analyzing the average MSE of LMMSE estimation with random regressors.
result Performance improvement depends on sufficient samples and model complexity.
The ratio of two probability densities can be used for solving various machine learning tasks such as covariate shift adaptation (importance sampling), outlier detection (likelihood-ratio test), and feature selection (mutual information). Recently, several methods of directly estimating the density ratio have been deve…
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more explicit relationship for the graph between knotting and linking, whi…
Researchers find optimal configurations of complex knots and links.
problem Finding the most efficient configurations of complex knots and links.
method Minimizing Möbius and Minimum Distance energies by describing them with a small number of free parameters.
result Optimal geometries for Hopf links, Borromean rings, and chain links are found.
We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the aperture. This allows us to use results about the invertibility of the kernel mean m…
Study on multiplicities in length spectrum of Salem numbers.
problem Understanding multiplicities in the length spectrum of Salem numbers.
method Analysis of square-rootable Salem numbers and their growth rate.
result Proved exponential growth rate for mean multiplicities in length spectrum.
Study on counting Salem numbers linked to geodesics in hyperbolic orbifolds.
problem Quantifying Salem numbers associated with closed geodesics in arithmetic hyperbolic orbifolds.
method Analytical and asymptotic methods to estimate the number of square-rootable Salem numbers.
result Found that non-compact arithmetic 3-dimensional orbifolds define square-rootable Salem numbers of degree 4.
The paper identifies saddlepoints in unsupervised auto-encoding neural nets.
problem The risk landscape of unsupervised least squares in auto-encoding neural nets.
method Established an equivalence between unsupervised least squares and principal manifolds, discussed regularization strategies for auto-encoders.
result All non-trivial critical points in auto-encoding are saddlepoints, which are degenerate in overcomplete auto-encoding.
A new method simulates square-root processes efficiently.
problem Simulating square-root processes accurately and efficiently.
method Simulate the integrated square-root process instead of the square-root process itself.
result High precision with low number of time steps, and exact limiting Inverse Gaussian distributions.
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, th…
We give a Conway-Gordon type formula for invariants of knots and links in a spatial complete four-partite graph in terms of the square of the linking number and the second coefficient of the Conway polynomial. As an application, we show that every rectilinear spatial contains a nontrivial Ha…
Researchers find a Steenrod square for link Floer homology.
problem Computing the second Steenrod square for link Floer homology.
method Explicitly framing moduli spaces and constructing a framed 1-flow category.
result An algorithm for computing the second Steenrod square for all grid homology versions.
In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruen…
A new mosaic system for immersed surface-links is introduced.
problem Constructing a mosaic system for immersed surface-links.
method Using singular marked graph diagrams, the mosaic number for immersed surface-links is defined and discussed.
result A mosaic system for immersed surface-links is established.
We investigate a class of hierarchical mixtures-of-experts (HME) models where exponential family regression models with generalized linear mean functions of the form psi(ga+fx^Tfgb) are mixed. Here psi(...) is the inverse link function. Suppose the true response y follows an exponential family regression model with mea…
Efficiently estimates private least squares with linear error growth.
problem Private estimation of ordinary least squares with bounded residuals and leverage.
method Scaled noise added to a stable nonprivate estimator of the regression vector.
result Near-optimal accuracy guarantee with linear error growth in dimension.
Improved stock volume prediction using Kalman Filters with various hidden states.
problem Improving accuracy of intraday trading volume prediction.
method Extended Kalman Filter with various hidden states for different stocks, using cross-validation to determine optimal state number.
result Demonstrated improved accuracy through comparison experiments and numerical analysis.
Algorithm finds mosaic numbers for knots with 10 or fewer crossings.
problem Efficiency in representing knots as mosaics.
method Algorithmic programming approach to find mosaic and tile numbers.
result Table of knot mosaics and mosaic number for prime knots with 10 or fewer crossings.
The paper constructs exotic surface links in 4-ball, proving their Brunnian nature.
problem Investigating exotic surface links in 4-ball.
method Two constructions of Brunnian exotic surface links, using satellite operations and covering links.
result Provided constructions of exotic Brunnian surface links with varied properties.
A fast method for LOOCV in k-NN regression reduces computation time.
problem Efficient computation of LOOCV for k-NN regression.
method Identical LOOCV estimate to (k+1)-NN MSE on training data.
result LOOCV computation can be done with (k+1)-NN regression once.
Compatible Relative Open Books on Relative Contact Pairs via Generalized Square Bridge Diagramsmath.GT
Article generalizes open book construction for 5D contact pairs.
problem Constructing compatible open books on relative contact pairs.
method Introduces generalized square bridge position for 5D Legendrian links.
result Algorithm constructs relative open book decompositions on relative contact pairs.
The estimation of probabilities of network edges from the observed adjacency matrix has important applications to predicting missing links and network denoising. It has usually been addressed by estimating the graphon, a function that determines the matrix of edge probabilities, but this is ill-defined without strong a…
ABae efficiently computes subset means with expensive predicates using stratified sampling.
problem Computing subset means with expensive predicates efficiently.
method Stratified sampling and proxy models.
result Mean squared error of when is split evenly between stages.
Arithmetic Kei Theorymath.NT
New invariant for square-free integers derived from kei theory.
problem Developing numerical invariants for square-free integers.
method Defining a kei for each square-free integer and calculating a coloring invariant.
result Conjecture and proof of asymptotic average order for coloring invariant.
In recent years, kernel density estimation has been exploited by computer scientists to model machine learning problems. The kernel density estimation based approaches are of interest due to the low time complexity of either O(n) or O(n*log(n)) for constructing a classifier, where n is the number of sampling instances.…
New algorithms estimate Jacobian matrices for large-scale machine learning.
problem Efficiently computing search directions for large nonlinear least squares.
method Exploit low-rank structure in Hessian to estimate Jacobian matrices.
result Two algorithms perform well compared to state-of-the-art methods.
Proposes a method to compute the second Steenrod square for odd Khovanov homology.
problem Computing the second Steenrod square for odd Khovanov homology.
method Proposes a new method to compute the second Steenrod square, showing it to be a link invariant.
result Shows the proposed method gives a refinement of the Rasmussen s-invariant with coefficients.
We develop an algorithm of polynomial time complexity to construct the Grushko decomposition of fundamental groups of graphs of free groups with cyclic edge groups. Our methods rely on analysing vertex links of certain CAT(0) square complexes naturally associated with a special class of the above groups. Our main resul…
Artificial neural network training with stochastic gradient descent can be destabilized by "bad batches" with high losses. This is often problematic for training with small batch sizes, high order loss functions or unstably high learning rates. To stabilize learning, we have developed adaptive learning rate clipping (A…
The aim of this paper is to propose distributed strategies for adaptive learning of signals defined over graphs. Assuming the graph signal to be bandlimited, the method enables distributed reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of sampled observatio…
The purpose of this paper is to study geometrically simply-connected homotopy 4-spheres by analyzing -component links with a Dehn surgery realizing . We call such links R-links. Our main result is that a homotopy 4-sphere that can be built without 1-handles and with only two 2-handles is diff…
We consider the problem of subspace estimation in a Bayesian setting. Since we are operating in the Grassmann manifold, the usual approach which consists of minimizing the mean square error (MSE) between the true subspace and its estimate may not be adequate as the MSE is not the natural metric in the Gra…
The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.
problem Counting geodesic paths in triangulations to infer topological invariants.
method Random walks on higher-dimensional skeletons of triangulations.
result Recovery of Betti numbers and linking numbers of manifolds.
The paper analyzes how data augmentation affects the test error in regression models.
problem Understanding the impact of data augmentation on the test error in regression models.
method Characterizes the test error in terms of population quantities and augmentation statistics.
result Provides a tight characterization of the test error in mean squared error.
Biased mean regression estimates factors exceeding expected loss or radiation release severity.
problem Estimating factors exceeding expected loss or radiation severity levels.
method Biased mean regression using superexpectation error minimization.
result Equivalent to quantile regression and CVaR optimization under specific conditions.
Binary data matrices can represent many types of data such as social networks, votes, or gene expression. In some cases, the analysis of binary matrices can be tackled with nonnegative matrix factorization (NMF), where the observed data matrix is approximated by the product of two smaller nonnegative matrices. In this …
We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings:…