arXiv research

HADES detects data singularities quickly and accurately.

We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to def…

Detects singularities in complex data to improve machine learning models.

The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.

Paper tackles singularity detection in PDEs using data-driven self-supervised learning.

Improves community detection in directed networks with theoretical guarantees.

A new algorithm detects out-of-distribution samples by concentrating them in feature space.

We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibilit…

If a given behavior of a multi-agent system restricts the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct…

New method detects essential tori in mixed singularity links.

TDA detects financial bubbles through early warning signals.

We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at $r = 0$. We show that on those hypersurfaces …

Paper detects duality obstruction in smooth calibrations.

Paper optimizes sparse feature selection for cancer detection using GSVP and SVM.

This paper is devoted to studying the structure of codimension one singular holomorphic foliations on $({\mathbb C}^3,0)$ without invariant germs of analytic surface. We focus on the so-called CH-foliations, that is, foliations without saddle nodes in two dimensional sections. Considering a reduction of singularities, …

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any co…

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in t…

Paper solves the minimal generating set problem for singular Reidemeister moves.

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular…

Exact recovery method for community detection in Gaussian mixtures with dependent noise.

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer homology and contact geometry. It uses open books; the contact invariants we defined in the instanton Floer setting; a bypass exact triangle in sutured instanton homology, proven here; and Kronheimer and Mrowka's spe…

The paper uses deep learning to detect financial market regimes from correlation matrices.

Wavelet analysis reveals limitations in detecting multifractality in signals with isolated singularities.

Randomized SVD shows phase transitions in noisy data.

A novel kernel-based test detects equality versus singularity of two probability measures.

We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically rigorous but highly idealized.

Introduces a new length functional for Ricci flow to detect steady solitons.

A new method for anomaly detection using random subspaces and Gaussian mixture models.

It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincaré Duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation betwee…

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…

Hypothesis testing in singular models is fundamentally about identifiable vs. non-identifiable parameters.

A new method detects anomalies in multivariate streams without unit dependence.

We present a general method to detect and extract from a finite time sample statistically meaningful correlations between input and output variables of large dimensionality. Our central result is derived from the theory of free random matrices, and gives an explicit expression for the interval where singular values are…

We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$. We tackle this problem by the so-called…

Consider a supervised dataset $D=[A\mid \textbf{b}]$, where $\textbf{b}$ is the outcome column, rows of $D$ correspond to observations, and columns of $A$ are the features of the dataset. A central problem in machine learning and pattern recognition is to select the most important features from $D$ to be able to predic…

Hierarchical organization is a cornerstone of complexity and multifractality constitutes its central quantifying concept. For model uniform cascades the corresponding singularity spectra are symmetric while those extracted from empirical data are often asymmetric. Using the selected time series representing such divers…

The paper analyzes how random perturbations affect RSVD and its applications.

New stability thresholds detect K-stability in Fano manifolds.

Unified framework for singular statistical models using observable charts.

Spectral methods are popular in detecting global structures in the given data that can be represented as a matrix. However when the data matrix is sparse or noisy, classic spectral methods usually fail to work, due to localization of eigenvectors (or singular vectors) induced by the sparsity or noise. In this work, we …

Paper extends cobordism maps in Khovanov and instanton homologies.

New method detects concept drift in data streams with missing values.

Study of null mean curvature flow on de Sitter lightcone, related to 2d-Ricci flow.

Paper analyzes singular subspace estimation in noisy matrix models.

New contact structures detected by contact homology.

Community detection is the task of detecting hidden communities from observed interactions. Guaranteed community detection has so far been mostly limited to models with non-overlapping communities such as the stochastic block model. In this paper, we remove this restriction, and provide guaranteed community detection f…

Bayesian neural networks can be simplified by parameterizing weights as rank-$r$ matrices, reducing parameter count and improving performance.