New complexity notion connects finite decomposition and asymptotic property C.
problem Understanding and connecting different properties in metric spaces.
method Introducing finite APC-decomposition complexity and proving its implications.
result Finite APC-decomposition complexity implies property A for metric spaces.
The aim of this paper is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this paper, three equivalent definitions for decomposition complexity are established. We prove that metric spaces with finite hyperbolic dimension have finite (weak) deco…
New upper bound for geodesic complexity derived from cut locus decompositions.
problem Understanding geodesic complexity in Riemannian manifolds.
method Study of decompositions of cut loci and their tangent fibers.
result Established a new upper bound for geodesic complexity.
We study the topological types of pants decompositions of a surface by associating to any pants decomposition P, in a natural way its pants decomposition graph, Γ(P). This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…
P-moves connect different 3-manifold decompositions.
problem Connecting different 3-manifold decompositions.
method Using P-complex and Morse 2-functions, we show P-moves between pants-block decompositions.
result Any two pants-block decompositions are related by a finite sequence of P-moves.
We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov's finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the perma…
VecHGrad solves complex tensor decomposition problems more accurately and efficiently.
problem Complex tensor decomposition with multiple matrices and diagonal tensors.
method VecHGrad algorithm using gradient, Hessian-vector product, and adaptive line search.
result VecHGrad converges faster and more accurately than existing methods.
New f-vectors reveal geometric Lefschetz-like decompositions of flag spheres.
problem Understanding f-vectors of balanced simplicial complexes and flag spheres. method Analyzing h-vectors and f-vectors of flag spheres and balanced simplicial complexes. result Found f-vectors leading to geometric Lefschetz-like decompositions. The paper analyzes solitonic components of toric manifolds.
problem Analyzing solitonic components of toric manifolds.
method Computing eigenfunctions of a solitonic complex Laplacian operator.
result Determination of solitonic decomposition of Fano toric manifolds.
Decomposition complexity for metric spaces was recently introduced by Guentner, Tessera, and Yu as a natural generalization of asymptotic dimension. We prove a vanishing result for the continuously controlled algebraic K-theory of bounded geometry metric spaces with finite decomposition complexity. This leads to a proo…
Algorithm constructs Grushko decomposition of certain groups.
problem Decomposing fundamental groups of graphs of free groups.
method Analyzing vertex links of CAT(0) square complexes.
result Transforms complex to one with strong connectivity vertex links.
Algorithm constructs JSJ decomposition for hyperbolic groups.
problem Constructing JSJ decompositions for hyperbolic groups.
method Combinatorial and geometric analysis of immersed cycles in CAT(0) square complexes.
result First algorithm with explicit time bound for JSJ decompositions.
In this paper, we introduce and study various kinds of decomposition complexity. First, we give a characterization of residually finite groups having finite decomposition complexity (FDC). Secondly, we introduce equi-variant straight FDC (sFDC), and prove that a group having equi-variant sFDC if and only if its box spa…
Study of quaternionic hyperbolic space bisectors and their decompositions.
problem Understanding bisectors in quaternionic hyperbolic geometry.
method Developed theory of quaternionic bisectors, showed various decompositions, derived projection formulas.
result Introduced fan decompositions of quaternionic bisectors by totally geodesic submanifolds isometric to complex hyperbolic space.
Proposes a faster Isomap algorithm by reducing eigenvalue decomposition complexity.
problem High computational complexity of Isomap, especially in eigenvalue decomposition stage.
method Introduces a projection operator to reduce the complexity of the eigenvalue decomposition stage to linear order.
result Reduces Isomap's computational complexity to linear order while preserving structural information.
Classifies CR submanifolds in complex hyperbolic spaces.
problem Understanding CR submanifolds in complex hyperbolic spaces.
method Classifying orbits of a subgroup of the solvable part of the Iwasawa decomposition.
result Classification of homogeneous CR submanifolds.
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure n-dimensional regular CW-complexes. result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.
problem Developing a Lp-Hodge decomposition on sub-Riemannian contact manifolds. method Using a Sobolev approach and recent results from [4] and [6].
result Established an Lp-Hodge decomposition theorem for Rumin's forms on sub-Riemannian contact manifolds. We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\m…
A generalized complex manifold which satisfies the ∂∂-lemma admits a Hodge decomposition in twisted cohomology. Using a Courant algebroid theoretic approach we study the behavior of the Hodge decomposition in smooth and holomorphic families of generalized complex manifolds. In particular we …
Formula for complex SVD backpropagation developed.
problem No specific problem stated; focuses on complex SVD.
method Back propagation formula for complex SVD developed.
result Back propagation formula for complex SVD created.
New criterion for almost-complex 4-manifolds using polyhedral decompositions.
problem Bounding the genus of surfaces in almost-complex 4-manifolds.
method Polyhedral decompositions and adjunction criterion.
result Established adjunction inequality for almost-complex 4-manifolds.
Enhanced loop space decomposition for specific Poincaré complexes.
problem Decomposing the loop space of certain high-dimensional complexes.
method Utilizing a result from BT2 to simplify and extend Beben and Wu's work.
result Improved understanding of the loop space structure of (2n−2)-connected (4n−1)-dimensional Poincaré Duality complexes. Concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index <=2 (``Heegard'' decompositions).Sample result: Two 2-complexes are (up to 2-deformation) dual spines of a Heegard decomposition of the 4-sphere if and only if they satisfy the conclusions of the Alexander-Lefshet…
A Levi-Malcev type decomposition for 2-step solvable Lie algebras with a complex structure
problem Decomposition of 2-step solvable Lie algebras with a complex structure method Proving a Levi-Malcev type decomposition
result Fino-Vezzoni conjecture holds for 2-step solvable unimodular Lie algebras We present a simple, general technique for reducing the sample complexity of matrix and tensor decomposition algorithms applied to distributions. We use the technique to give a polynomial-time algorithm for standard ICA with sample complexity nearly linear in the dimension, thereby improving substantially on previous b…
Smooth 4-manifolds have simple horizontal decompositions.
problem Classifying smooth, closed, orientable 4-manifolds.
method Horizontal handlebody decomposition.
result Simplest horizontal decompositions classify closed 4-manifolds.
Let J1 be the real form of a complex simple Jordan algebra such that the automorphism group is F4(−20). By using some orbit types of F4(−20) on J1, for F4(−20), explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's Kε−Iwasawa decomp…
The study refines contingency matrices for complex stratification and braid group cohomology.
problem Combinatorics of contingency matrices and their applications.
method Refinement of complex stratification and study of braid group cohomology.
result Totally positive meta-matrix formed by contingency matrix sizes.
We analyzed SVD and variants for eigenpair computation, comparing their time and space complexities.
problem Comparing time and space complexities of SVD and variants for eigenpair computation.
method Comparison of SVD, truncated SVD, Krylov method, and Randomized PCA in terms of time and space complexity.
result Krylov method and Randomized PCA perform well only when k << n.
Classifies 4-manifolds with genus one horizontal handlebody decomposition.
problem Classifying 4-manifolds with specific properties.
method Classification using Euler characteristic and handlebody decomposition.
result Found a large family of rational homology balls that embed into CP2. The Hodge theorem connects cohomology groups on compact Kähler manifolds.
problem Establishing a relationship between cohomology groups on compact Kähler manifolds.
method Proving the Hodge decomposition theorem on compact d-Kähler manifolds.
result Hodge decomposition theorem on compact d-Kähler manifolds.
We study properties concerning decomposition in cohomology by means of generalized-complex structures. This notion includes the C∞-pure-and-fullness introduced by Li and Zhang in the complex case and the Hard Lefschetz Condition in the symplectic case. Explicit examples on the moduli space of the Iwas…
Constructs a simplicial cell decomposition of complex projective space for n ≥ 2.
problem Finding a simplicial cell decomposition for complex projective space.
method Starting with a standard crystallisation of the 2-sphere, constructing a simplicial subdivision, and quotienting by the Sym(n) action.
result Explicit construction of a simplicial cell decomposition of complex projective space for n ≥ 2.
Study of Riemannian foliations with bounded geometry, proving leafwise Hodge decomposition.
problem Understanding Riemannian foliations with bounded geometry.
method Leafwise Hodge decomposition and associated smoothing operators.
result Extension of Novikov differential complex to leafwise version.
Study on Hodge decompositions for Lie algebroids on manifolds with boundary.
problem When does the Chevalley-Eilenberg differential admit a Hodge decomposition?
method Introduce concepts like Cauchy-Riemann structures, elliptic and non-elliptic boundary points, q-convexity, and use them to prove Hodge decompositions.
result Hodge decompositions for q-convex elliptic Lie algebroids on manifolds with boundary.
We formalize the concept of a family of metric spaces satisfying a coarse property uniformly and we generalize finite decomposition complexity of Erik Guentner, Romain Tessera, and Guoliang Yu. Of particular interest are results determining sufficient conditions for a metric space to satisfy Property A of Guoliang Yu.
In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be ob…
The paper studies complex affine structures near irregular singularities.
problem Understanding complex affine structures near irregular singularities.
method Introducing local invariants and a Delaunay decomposition.
result Upper bounds on the complexity of the Delaunay decomposition.
Identifying important components or factors in large amounts of noisy data is a key problem in machine learning and data mining. Motivated by a pattern decomposition problem in materials discovery, aimed at discovering new materials for renewable energy, e.g. for fuel and solar cells, we introduce CombiFD, a framework …
Invariants for 3-manifolds with toral boundaries, related by sutured decompositions.
problem Invariants for 3-manifolds with toral boundaries and non-degenerate Thurston norm.
method Constructing an invariant called guts and proving its invariance under sutured decompositions.
result The guts of different homology classes are related by sutured decompositions.
A new method reduces model complexity in DMD using LARS.
problem Building accurate reduced-order models from data.
method Least Angle Regression (LARS) for Dynamic Mode Decomposition (DMD).
result LARS4DMD produces comparable performance to DMDSP with less complexity.
A novel approach to improve knowledge base completion using tensor decomposition.
problem Knowledge Base Completion (KBC) as a tensor completion problem.
method Canonical Tensor Decomposition (CP) with novel regularizers and reformulation.
result Improved KBC results using CP decomposition and ComplEx model.
New symplectic caps and embeddings found in complex projective plane.
problem Embeddings of homology balls in complex projective plane.
method Handlebody construction of symplectic caps and embeddings.
result First examples of symplectic handlebody decompositions of a closed symplectic 4-manifold.
Paper addresses statistical efficiency and scalability in tensor train decomposition.
problem Statistical inefficiency and scalability issues in tensor train decomposition.
method Introduces a convex relaxation and alternating optimization method with randomization.
result Derives error bounds and demonstrates method's performance on real data.
New algorithm learns halfspaces with noise using Forster decomposition.
problem Learning halfspaces in noisy data.
method Forster decomposition and efficient mixture of distributions.
result First polynomial-time algorithm with strongly polynomial sample complexity.
The paper studies posets from decompositions in symmetric monoidal categories.
problem Understanding posets from decompositions in symmetric monoidal categories.
method Defining decompositions and partial decompositions, complexes of frames, partial bases, and ordered versions.
result Unified approach to combinatorics and homotopy type of posets and complexes.