Stability in cohomology of linear subspace arrangements.
problem Stability in cohomology of linear subspace arrangements.
method FI-modules, representation stability, generalized character polynomials.
result Cohomology of linear subspace arrangements satisfies strong form of representation stability.
The paper explores how the cohomology of certain space arrangements stabilizes as the number of subspaces increases.
problem Stability of cohomology groups of complements of linear subspace arrangements.
method Representation stability in the context of cohomology groups, focusing on arrangements invariant under permutation of coordinates.
result Bounds on stabilization and alternative proof for the stabilization of cohomology groups.
The paper improves conditions for unique recovery in homomorphic sensing of subspaces.
problem Unique recovery of points in a linear subspace from their images under linear maps.
method Tighter and simpler conditions for unique recovery in single and subspace arrangement cases, extending to noise stability.
result Conditions for unique recovery in homomorphic sensing are improved and unified.
We study the Dictionary Learning (aka Sparse Coding) problem of obtaining a sparse representation of data points, by learning \emph{dictionary vectors} upon which the data points can be written as sparse linear combinations. We view this problem from a geometry perspective as the spanning set of a subspace arrangement,…
A new framework for graph representation learning.
problem Acquiring continuous representations of discrete objects like graphs.
method Nested SubSpace (NSS) arrangement and Disk-ANChor ARrangement (DANCAR).
result Successfully embedded WordNet in 20-dimensional space with high F1 score.
The paper establishes a Miyaoka-Yau type inequality for hyperplane arrangements in complex projective space.
problem Finding a lower bound for the total sum of multiplicities of codimension 2 intersection subspaces of a hyperplane arrangement.
method Using a quadratic form defined by the intersection poset of the hyperplane arrangement, and applying the Bogomolov-Gieseker inequality for parabolic bundles.
result The inequality Q(a,…,a)≤0 gives a lower bound for the total sum of multiplicities of codimension 2 intersection subspaces of the hyperplane arrangement, with equality conditions provided. We provide abelianizations of differentiable actions of finite groups on smooth real manifolds. De Concini-Procesi wonderful models for (local) subspace arrangements and a careful analysis of linear actions on real vector spaces are at the core of our construction. In fact, we show that our abelianizations have stabili…
The k-th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the complex algebraic n-torus. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V_k…
Geometric arguments show simplicial arrangements with few double points can't have an irreducible cubic curve dual.
problem Classifying simplicial arrangements with a linear bound on double points.
method Geometric arguments and structure theorem from Green and Tao.
result Simplicial arrangements with few double points can't have an irreducible cubic curve dual.
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-…
Study shows certain toric arrangements have minimal topological complements.
problem Understanding the topology of toric arrangements.
method Associated a matrix to toric arrangements and analyzed those with maximal rank.
result Complement manifolds of these toric arrangements are diffeomorphic to centered ones.
Paper presents a new method for multiclass classification using hyperplane arrangements.
problem Developing efficient multiclass classifiers.
method Mixed integer programming formulations with hyperplane arrangements, kernel trick adaptation, and dimensionality reductions.
result Our proposal outperforms other methods in multiclass classification tasks.
We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of n hyperplanes in an r-dimensional linear space is min{n+1,2r}.
This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion problem to situations in which the matrix rank can be quite high or even full r…
Existence of smooth valuations on subspaces is shown for certain conditions.
problem Existence of smooth valuations on subspaces with given restrictions.
method Analyzing compatibility and using recursive descriptions of the cosine transform.
result Compatibility is sufficient for extensibility in certain regimes.
Study stabilizes arithmetic statistics of rational maps over finite fields.
problem Stability of arithmetic statistics of rational maps over finite fields.
method Representation stability and arithmetic statistics of spaces of 0-cycles.
result Arithmetic quantities associated to rational maps over finite fields stabilize as degree increases.
High-dimensional data often lie in low-dimensional subspaces corresponding to different classes they belong to. Finding sparse representations of data points in a dictionary built using the collection of data helps to uncover low-dimensional subspaces and address problems such as clustering, classification, subset sele…
We study an elementary problem of the topological robotics: collective motion of a set of n distinct particles which one has to move from an initial configuration to a final configuration, with the requirement that no collisions occur in the process of motion. The ultimate goal is to construct an algorithm which will…
Conditions for polyhedral Kähler metrics on CP^n with specific singularities.
problem Existence of polyhedral Kähler metrics on complex projective space with specified singularities.
method Parabolic Kobayashi-Hitchin correspondence, linear and quadratic constraints on cone angles.
result Necessary and sufficient conditions for the existence of polyhedral Kähler metrics on CP^n.
Upper bounds on fixed points in PWL neural networks with hyperplane analysis.
problem Analyzing the number of fixed points in neural networks with PWL activation.
method Hyperplane arrangements to bound the number of fixed points.
result Upper bounds on the number of fixed points for PWL networks, showing exponential growth in layers.
In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we…
The study finds PK cone metrics on complex manifolds near hyperplane arrangements.
problem Finding metrics on complex manifolds near singularities.
method Analyzing flat torsion-free meromorphic connections on \(\mathbb{C}^n\) with simple poles at hyperplanes.
result Metric completion of certain connections yields PK cone metrics on \(\mathbb{C}^n\).
For an arrangement of n pseudolines in the real projective plane let us denote by ti the number of vertices incident to i lines. We obtain a linear on ti inequality similar to the Hirzebruch one, but with an elementary proof. We present an algorithm for producing lower bounds of the number of regions basing o…
Kernel models learn low-dimensional predictive subspaces from input data.
problem Learning effective feature transformations in kernel models.
method Study of a compositional kernel ridge regression model.
result Global minimizers of the objective function identify the subspace with high probability.
Investigates projections onto explicit subspaces and their variance effects.
problem Understanding the variance preservation in explicit subspace projections.
method Investigates projections onto explicit subspaces of varying dimensionality and analyzes the variance effects.
result Developed new bounds for Euclidean distances and inner products.
The affine Grassmannian generalizes Euclidean and linear subspaces with rich geometric properties.
problem Formulating machine learning and statistical problems on the affine Grassmannian.
method Showed the affine Grassmannian has multiple structures and affords an analogue of Schubert calculus.
result The affine Grassmannian serves as a concrete computational platform for various machine learning and statistical problems.
Paper shows affine constraint is unnecessary for high-dimensional data.
problem The necessity of an affine constraint in affine subspace clustering.
method Theoretical and empirical analysis of conditions for correctness of affine subspace clustering methods.
result Affine constraint has negligible effect on clustering performance for high-dimensional data.
Real line arrangements with 3n lines intersect each other in n+1 points are related to finite complex reflection groups.
problem Identifying all real line arrangements in CP^2 with the Hirzebruch property.
method Analyzing the intersections of lines and confirming the relationship with finite complex reflection groups.
result There exist exactly four real line arrangements in CP^2 satisfying the Hirzebruch property.
A new method learns multiple subspaces from data.
problem Learning discriminative representations for data on multiple subspaces.
method Sequential game using CTRL framework to solve linear subspaces.
result Equilibrium solutions provide correct representations.
Euclidean systems and real PK arrangements linked via geometry.
problem Establishing a connection between Euclidean systems and real PK arrangements.
method Proving a correspondence between Euclidean ∨-systems and real PK arrangements, and showing homeomorphism of moduli spaces. result Moduli space of Euclidean ∨-systems is homeomorphic to a polytope's interior, and hyperplane arrangements are simplicial. We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…
Study non-asymptotic bounds on correlation in high-dimensional linear systems, revealing invariant subspaces and bottlenecks.
problem Understanding correlation and mixing in high-dimensional linear systems with Gaussian noise.
method Sampling from sub-trajectories, using Talagrand's inequality, and analyzing invariant subspaces.
result Large discrepancy between algebraic and geometric multiplicity leads to bottlenecks between invariant subspaces.
New infinite K(π,1) arrangements found in higher dimensions.
problem Finding new infinite K(π,1) arrangements in higher dimensions. method Endowing Falk complexes with an injective metric.
result New examples of infinite K(π,1) arrangements in dimension n>2. This work presents GROUSE (Grassmanian Rank-One Update Subspace Estimation), an efficient online algorithm for tracking subspaces from highly incomplete observations. GROUSE requires only basic linear algebraic manipulations at each iteration, and each subspace update can be performed in linear time in the dimension of…
The Sample Compression Conjecture of Littlestone & Warmuth has remained unsolved for over two decades. This paper presents a systematic geometric investigation of the compression of finite maximum concept classes. Simple arrangements of hyperplanes in Hyperbolic space, and Piecewise-Linear hyperplane arrangements, are …
The regular \Z^r-covers of a finite cell complex X are parameterized by the Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording when the Betti numbers b_1,..., b_i of the corresponding covers are finite carves out certain subsets Ω^i_r(X) of the Grassmannian. We present here a method, essent…
Study shows asphericity isn't inherited in certain arrangements.
problem Asphericity in hyperplane arrangements is not always inherited.
method Examples of K(π,1)-arrangements with restricted asphericity. result Asphericity is not hereditary among hyperplane arrangements.
Icosidodecahedral arrangement has torsions in homology, and is a K(π,1).
problem Identifying arrangements with torsions in homology.
method Proving the icosidodecahedral arrangement is a K(π,1).
result The icosidodecahedral arrangement is a K(π,1).
Agents collaborate to reduce regret in a multi-agent linear bandit problem with side information.
problem Reducing regret in a multi-agent stochastic linear bandit with side information.
method A decentralized algorithm where agents communicate subspace indices and each plays a projected LinUCB on the corresponding low-dimensional subspace.
result Per-agent finite-time regret is much smaller when agents communicate compared to non-communicating case.
Autoencoders reveal principal component subspaces.
problem Recovering principal component loadings from autoencoder weights.
method Using linear autoencoders with specific configurations.
result Autoencoder weights span the same subspace as principal component loadings.
Paper proves linear convergence of SCMS algorithm for directional data.
problem Identifying density ridges in directional data.
method Generalized SCMS algorithm to directional data, derived from SCGA with adaptive step size.
result Linear convergence of the proposed directional SCMS algorithm.
This paper studies topological properties of line arrangements in complex projective plane.
problem Understanding the topology of line arrangements in complex projective plane.
method Established foundational results using homological methods to compute cohomology rings and studied homotopy types.
result Complement of a symplectic line arrangement has the homotopy type of a minimal CW complex.
Study a specific line arrangement and compute its fundamental group via braid monodromy.
problem Compute the fundamental group of a specific line arrangement's complement.
method Use braid monodromy to compute the fundamental group.
result The resulting presentation of the fundamental group coincides with the modified Artin presentation.
Research examines arrangements of hyperplanes in real projective spaces, focusing on specific cases.
problem Analyzing the structure of hyperplane arrangements in real projective spaces.
method Investigates arrangements of m hyperplanes in the n-dimensional real projective space, with a focus on m=n+3 and n=3 or n=4. result Provides insights into the structure of chambers cut out by these specific hyperplane arrangements.
Efficiently clusters large datasets with a subset of landmarks.
problem High computational complexity in subspace clustering for large-scale datasets.
method Selects a subset of landmarks to reduce the clustering problem to linear time.
result Subspace clustering method runs in linear time with respect to the size of the original data.
GOTabPFN improves tabular model performance with compact tokenization for HDLSS data.
problem Making tabular models effective for high-dimensional, low-sample size data without retraining.
method Introducing Graph-guided Ordering with Local Refinement (GO-LR) and Neuro-Inspired Subunit Compression (NSC) to create compact meta-features.
result GOTabPFN improves stability and accuracy in tabular benchmarks with compact tokenization.
Study restricts line arrangements with odd points using topological arguments.
problem Restrictions on line arrangements with singular points of odd multiplicity.
method Topological arguments on locally-flat spheres in 4-manifolds.
result No line arrangement with 13 lines and only triple points exists.