Random Intersection Chains selects important interactions from categorical features.
problem Heavy computational burden in considering all interactions for categorical features.
method Randomly generates chains of intersections, estimates and selects frequent patterns.
result Selected patterns are the most frequent in the data set.
Computes expected number of real intersection points of essential variety with random linear spaces.
problem Computing the expected number of real intersection points of the essential variety with random linear spaces.
method Two probability distributions for linear spaces: invariant under orthogonal group action and one motivated from computer vision. Used Monte Carlo simulation for the latter.
result Expected number of real intersection points lies in the interval (3.95 - 0.05, 3.95 + 0.05) with high probability.
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distrib…
Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class…
The paper extends Busemann's inequalities to complex and quaternionic spaces.
problem Extending Busemann's inequalities to complex and quaternionic vector spaces.
method Proof leverages a monotonicity property under symmetrization with respect to complex or quaternionic hyperplanes.
result Standard Steiner symmetrization does not exhibit the monotonicity property in complex or quaternionic spaces.
We extend Kac-Rice formula to compute expected intersections of random submanifolds.
problem Computing expected intersections of random submanifolds.
method Generalized Kac-Rice formula using measure theory and integration.
result Formula computes expected cardinality of preimages of submanifolds via random maps.
The paper calculates large genus limits for quadratic differential volumes and constants.
problem Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials.
method Combining recursive relations (Virasoro constraints) and asymmetric simple random walk jump probabilities.
result Confirm predictions about Masur-Veech volumes and area Siegel-Veech constants.
SBT model uses randomized sharding and sub-models to improve Bayesian Additive Regression Trees.
problem Improving efficiency and accuracy of Bayesian Additive Regression Trees.
method Randomized sharding, sub-models, intersection tree structure, optimal design.
result Theoretical optimal weights and worst-case complexity of SBT model.
BEGIN network models binary data without parametric assumptions.
problem Conditional independence in non-parametric families of binary data.
method BEGIN network models binary data using sparse linear representations and block factorizations.
result BEGIN network captures conditional independence for arbitrary binary and multinomial variables.
Random curves on surfaces have predictable properties as they grow.
problem Characterizing topological properties of random curves on surfaces.
method Analyzing a simple random walk on the Cayley graph of surface groups.
result The properties of random curves are generic and predictable as they grow.
This work investigates the intersection property of conditional independence. It states that for random variables A,B,C and X we have that X independent of A given B,C and X independent of B given A,C implies X independent of (A,B) given C. Under the assumption that the joint distribution has a co…
Probabilistic theory counts intersections in Riemannian spaces.
problem Counting intersections in Riemannian homogeneous spaces.
method Introduces probabilistic intersection ring HE(M), a graded commutative and associative real Banach algebra. result Probabilistic intersection ring structure defined for spheres, real projective space, and complex projective space.
Study of random multicurves and square-tiled surfaces on large genus surfaces.
problem Understanding the geometry and combinatorial properties of random multicurves and square-tiled surfaces on surfaces of large genus.
method Combination of combinatorial and geometric analysis, including large genus asymptotic analysis of moduli space volumes and intersection numbers.
result Random multicurves and square-tiled surfaces have well-approximated properties by random permutations, with specific expected values.
This paper offers a new algebraic perspective of GCCA using subspace intersection.
problem Finding common variables across multiple feature representations.
method Subspace intersection approach based on a (bi-)linear generative model.
result GCCA is equivalent to subspace intersection, with conditions for identifiable common subspace.
The Hanna Neumann conjecture gives a bound on the intersection of finitely generated subgroups of free groups. We explore a natural extension of this result, which turns out to be true only in the finite index case, and provide counterexamples for the general case. We also see that the graph-based method of generating …
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
Method detects interactions for better CTR prediction.
problem Predicting click-through rate with high-dimensional categorical features and time-varying interactions.
method Online Random Intersection Chains (ORIC) for detecting informative interactions.
result ORIC detects high-interpretability interactions that improve CTR prediction.
Counting meanders on surfaces of arbitrary genus, with precise asymptotics.
problem Counting and understanding meanders on surfaces of arbitrary genus.
method Square-tiled surfaces, moduli spaces of Abelian and quadratic differentials, Witten-Kontsevich 2-correlators.
result Asymptotic probability and polynomial growth of meanders with intersections.
New lower bounds show learning intersections of halfspaces is hard even for a few halfspaces.
problem Learning intersections of halfspaces in polynomial time under standard assumptions.
method Unified connection to parallel pancakes distribution for proving hardness.
result Learning ω(loglogN) halfspaces in dimension N requires super-polynomial time under standard assumptions. In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be e…
Algorithm learns CNF formulas from random solutions under specific conditions.
problem Learning a CNF formula from uniform random solutions.
method Revisits Valiant's algorithm and applies Lovász local lemma conditions.
result Significantly reduces sample complexity for learning CNFs.
Extracts the finest pattern of mutual independence from data.
problem Inferring the finest mutual independence pattern from data.
method Estimate the set of valid patterns of dichotomic independence and use their intersection to infer the finest pattern.
result The method can estimate the finest mutual independence pattern from i.i.d. realizations of a multivariate normal distribution.
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first 3g−3 geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short (<log(g)) geodesics on a …
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algorithms in order to increase their convergence speed. We consider algorithms that sample from a probability density conditioned on a manifold…
We show an efficient algorithm for the following problem: Given uniformly random points from an arbitrary n-dimensional simplex, estimate the simplex. The size of the sample and the number of arithmetic operations of our algorithm are polynomial in n. This answers a question of Frieze, Jerrum and Kannan [FJK]. Our resu…
Geometric approach clusters intersecting manifolds with high probability.
problem Clustering intersecting d-dimensional manifolds.
method Compute locality graph on d-simplices using dihedral angles, then compute LAPD to separate manifold components.
result The method separates manifold components with high probability under random sampling.
Fairness audits fail under missing protected labels, especially at zero access.
problem Understanding the reliability of fairness audits with incomplete protected-label data.
method Introduced a seed-calibrated stress test to separate missingness effects from seed-to-seed movement.
result Missing protected labels do not significantly alter fairness mitigation methods, but they can lead to harmful intersectional outcomes.
New sparsification theorem for Gaussian processes reduces dimensionality.
problem Dimension-independent sparsification of Gaussian process suprema.
method Dimension-independent sparsification of Gaussian process suprema.
result Sparsifier size is independent of ∣T∣ and n. Finding interactions between variables in large and high-dimensional datasets is often a serious computational challenge. Most approaches build up interaction sets incrementally, adding variables in a greedy fashion. The drawback is that potentially informative high-order interactions may be overlooked. Here, we propos…
We investigate some geometric properties of the real algebraic variety Δ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in Δ. We exhibit conne…
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae …
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
3-manifold triangulation can be reconstructed from its intersection matrix.
problem Reconstructing the triangulation of 3-manifolds from their intersection matrix.
method Using the intersection matrix of a simplicial complex to determine the triangulation of a 3-manifold up to isomorphism.
result The intersection matrix is sufficient to determine the triangulation of a 3-manifold up to isomorphism.
We investigate the dynamics of 2-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space B of such semigroups. We show that for a parameter h in the intersection of B, the hyperbolicity locus ${\c…
We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call "homotopic curve shortening" (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂R2 of point obstacles, and evolves in discrete…
Randomized Geometric Algebra for Convex Neural Networks Optimizes Transfer Learning.
problem Training neural networks to global optimality via convex optimization.
method Randomized algorithms in Clifford's Geometric Algebra for hypercomplex vector spaces.
result Convex optimization and geometric algebra improve LLMs' robustness and reliability in transfer learning.
It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones fun…
The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…
Recent seminal work at the intersection of deep neural networks practice and random matrix theory has linked the convergence speed and robustness of these networks with the combination of random weight initialization and nonlinear activation function in use. Building on those principles, we introduce a process to trans…
We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential ste…
Conditions for curves on a torus with specific pairwise intersections.
problem Finding curves on a torus with prescribed pairwise intersections.
method Necessary and sufficient conditions for curves on a torus with given pairwise intersections.
result Necessary and sufficient conditions for the existence of curves on a torus with specific pairwise intersections.
Study self-intersections of arcs on a pair of pants, proving natural number spectrum.
problem Understanding self-intersections of arcs on a pair of pants.
method Algorithm to compute self-intersection number, bounds established in terms of word length.
result Spectrum of self-intersection numbers covers all natural numbers.
Virtual knots with same writhe polynomial have equivalent intersection graphs.
problem Equivalence of intersection graphs for virtual knots.
method Proved equivalence through writhe polynomial.
result Intersection graphs of virtual knots with the same writhe polynomial are equivalent.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
Estimates intersection pairing in hyperbolic 4-manifolds.
problem Estimating intersection pairing in hyperbolic 4-manifolds.
method Using Thurston norms of homology classes.
result Proved an estimate on intersection pairing.