arXiv research

Fiedler regularization uses spectral graph theory to improve neural network performance.

Fiedler regularization uses graph sparsity to improve neural network training.

In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided knots. I introduce the Kauffman bracket in the solid torus. Its Taylor expansion…

It is shown how Fiedler's `small state-sum' invariant for a braid can be calculated from the 2-variable Alexander polynomial of the link which consists of the closed braid together with the braid axis.

Crowdsourcing platforms are now extensively used for conducting subjective pairwise comparison studies. In this setting, a pairwise comparison dataset is typically gathered via random sampling, either \emph{with} or \emph{without} replacement. In this paper, we use tools from random graph theory to analyze these two ra…

We argue that the standard graph Laplacian is preferable for spectral partitioning of signed graphs compared to the signed Laplacian. Simple examples demonstrate that partitioning based on signs of components of the leading eigenvectors of the signed Laplacian may be meaningless, in contrast to partitioning based on th…

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.

The paper examines topological features of ReLU networks and their relation to decision boundaries and training loss.

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants of a knot in an $R^1$-fibration over a surface $F$. They take values in the free…

This text is intended to become in the long run Chapter 3 of our long saga dedicated to Riemann, Ahlfors and Rohlin. Yet, as its contents evolved as mostly independent (due to our inaptitude to interconnect both trends as strongly as we wished), it seemed preferable to publish it separately. More factually, our account…

Generative models for graphs have been typically committed to strong prior assumptions concerning the form of the modeled distributions. Moreover, the vast majority of currently available models are either only suitable for characterizing some particular network properties (such as degree distribution or clustering coe…

Fiedler and Mallet-Paret prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapuno…

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic an…

Develops a new method to recover large latent tree models efficiently.

Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to t…

The paper compares eigenvalues of Dirichlet, Neumann, and Laplacian on graphs.

New insights into spectral clustering reveal strong connections within eigenvectors.

Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of …

Transformer models show distinct spectral fingerprints under voice changes.

New nodal domain theorems for symmetric matrices via signed graphs.

Many modern datasets can be represented as graphs and hence spectral decompositions such as graph principal component analysis (PCA) can be useful. Distinct from previous graph decomposition approaches based on subspace projection of a single topological feature, e.g., the Fiedler vector of centered graph adjacency mat…

Study proves consistency of spectral clustering on hierarchical networks.

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple graphs for which every unit sphere is a d-sphere, the zero locus of (f-c) is a …

GNNRank uses neural networks to learn global rankings from competition match data.

Spectral dimensionality reduction methods enable linear separations of complex data with high-dimensional features in a reduced space. However, these methods do not always give the desired results due to irregularities or uncertainties of the data. Thus, we consider aggressively modifying the scales of the features to …

Improved spectral clustering via Gromov-Wasserstein Learning.

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair $(G,p)$ of graph $G$ together with a map $p$ of the vertices of $G$ into the Euclidean pla…

New method calculates Shapley values for uncertain functions.

New set-valued star-shaped risk measures introduced for better risk assessment.

The paper introduces Absolute Shapley Value to handle negative contributions in machine learning model training.

Study on 2-valued dynamics on complex plane, showing some dynamics can't be group actions.

Formula for Z_2-valued index of symmetric operators on manifolds.

New method converts p-values to e-values for more efficient CP and aggregation.

Introduces joint Shapley values to measure feature importance in models.

Proposes a low-cost method to set hyperparameters using optimized default values.

The paper introduces the Banzhaf value for robust data valuation in machine learning, addressing stochastic model performance.

A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors followed by (2) taking the absolute value of every entry…

Complex-valued neural networks are not a new concept, however, the use of real-valued models has often been favoured over complex-valued models due to difficulties in training and performance. When comparing real-valued versus complex-valued neural networks, existing literature often ignores the number of parameters, r…

We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.

The paper defines and analyzes set-valued stochastic integrals for Lévy processes.

Shapley value improves model interpretation but not causal inference.

This paper proposes a new approach to RL by focusing on the value-improvement path.

UA-LQE improves value function learning by selectively erasing uncertain entries in Q-matrix.

RDIS fills missing values in time series data explicitly.

E-values enhance conformal prediction methods.

Paper introduces a new method for classifying interval-valued time series.

Developing an explainable outlier detection method for interval-valued data using Shapley value-based approach.

Paper introduces v-CMC linking causality and utility.