New GPs model edge functions on complex networks, capturing divergence and curl.
problem Modeling flow data on networks with independent learning of Hodge components.
method Developed Hodge-compositional edge GPs using Hodge decomposition.
result Hodge-compositional edge GPs can represent any edge function and capture flow relevance.
DecompKAN improves time series forecasting accuracy and transparency.
problem Accurate and transparent time series forecasting in scientific domains.
method Combines decomposition, patching, normalization, and B-spline KAN edge functions.
result Achieves best or tied-best MSE on 20 of 36 comparisons across 9 datasets.
Temporal Functional Circuits explain KAN forecasts with interpretable edge functions.
problem Lack of mechanistic explanations in KAN forecasting.
method Transform KAN edge functions into faithful, temporally grounded explanations using a gated residual KAN.
result Gated KAN achieves lower MSE than linear-only models on regime-switching signals.
POSCMs extend SCMs for causal modeling with latent contexts.
problem Causal modeling with latent contexts and endogenous mechanisms.
method Kolmogorov-Arnold-Sprecher edge-functional decomposition for explicit parametrization.
result Identifiability of structure and mechanisms under latent context.
KANs replace fixed MLP weights with learnable edge functions, improving accuracy and interpretability.
problem Lack of interpretability and scalability in MLPs.
method KANs use learnable activation functions on edges instead of fixed weights, replacing weights with spline functions.
result KANs outperform MLPs in accuracy and interpretability with smaller models.
Given data sampled from a number of variables, one is often interested in the underlying causal relationships in the form of a directed acyclic graph. In the general case, without interventions on some of the variables it is only possible to identify the graph up to its Markov equivalence class. However, in some situat…
Adaptive RBF-KAN improves KANs by dynamically adjusting kernel parameters.
problem Efficiently approximating multivariate functions using learnable univariate edge functions.
method Integrates LOOCV-based kernel scale estimation with adaptive kernel learning.
result Adaptive RBF-KAN outperforms fixed kernel KANs on various benchmark functions.
Bayesian KANs achieve near-minimax posterior contraction rates in anisotropic Besov spaces.
problem Statistical foundation for Bayesian Kolmogorov-Arnold networks in anisotropic Besov spaces.
method Sparse Bayesian KANs with spike-and-slab priors, hyperprior on model size, and approximation complexity bounds.
result Posterior contraction rates depend on intrinsic anisotropic smoothness and effective dimension of the compositional structure.
Researchers describe and compare decompositions of Poincaré duality pairs.
problem Understanding and comparing different decompositions of Poincaré duality pairs.
method Developed and described edge splittings of decompositions based on group properties.
result Compared decompositions with two other related decompositions.
The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.
problem Decompositions of moduli spaces of vector bundles with fixed determinant of odd degree.
method Semiorthogonal decompositions, Grothendieck ring of varieties, mirror symmetry, graph potentials, Fukaya category.
result Evidence for a conjectural semiorthogonal decomposition of moduli spaces of rank 2 bundles with odd determinant.
The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.
problem Classifying decompositions of 3-manifolds with handlebodies.
method Studied decompositions of 3-sphere and lens spaces with three handlebodies, using stabilizations.
result Determined whether decompositions are stabilized.
We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition comp…
This paper generalizes octahedral decomposition to links in thickened surfaces.
problem Understanding the geometry of links in thickened surfaces.
method Octahedral decomposition of links in thickened surfaces.
result Nonpositive curvature of the complement and essential-ness of edges proved.
Researchers compute Goeritz groups for all (1,1)-link decompositions.
problem Computing Goeritz groups for all (1,1)-link decompositions.
method Analyzing surface decompositions and isotopy classes of homeomorphisms.
result Computed Goeritz groups for all (1,1)-link decompositions.
Study concordance of decompositions from defining sequences in 3-sphere.
problem Understanding concordance and bordism of decompositions from defining sequences.
method Relate to invariants of toroidal decompositions and cobordism of homology manifolds.
result At least uncountably many concordance classes of decompositions in 3-sphere.
Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…
Study shows OAT decomposition generates unexplained profit and loss, while SU decompositions depend on risk factor order.
problem Understanding profit and loss attribution in financial markets.
method Used financial market data from 2003 to 2022 to compare OAT, SU, and ASU decompositions.
result SU decompositions are sensitive to risk factor order and cannot identify all relevant risk factors.
A new algorithm speeds up CP decomposition for large tensors.
problem Efficiently processing large-scale tensors in real-time.
method Randomized online CP decomposition (ROCP) algorithm.
result ROCP reduces computing time and memory usage significantly.
Paper characterizes optimization landscape of Tucker decomposition.
problem Finding exact Tucker decomposition is a nonconvex optimization problem.
method Characterized the optimization landscape and provided a local search algorithm.
result All local minima are globally optimal if tensor has an exact Tucker decomposition.
A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…
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…
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…
New method uses random decompositions for high-dimensional Bayesian optimization.
problem Learning accurate decompositions for high-dimensional black-box functions.
method Data-independent random tree-based decomposition sampling.
result Random decomposition upper-confidence bound algorithm (RDUCB) yields significant empirical gains.
New varifold example shows decomposition failure.
problem Curvature varifolds cannot always be decomposed.
method Constructed a specific curvature varifold.
result Found a varifold with a non-preserved weak second fundamental form under decomposition.
Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
problem Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
method Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
result Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.
Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be k-manifolds (k=1,2), which generalize characterizations in the codimens…
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). Paper introduces a new principle for fair redistribution of insurance surplus.
problem Fair redistribution of surplus in life insurance policies.
method Introduces ISU decomposition principle based on infinitesimal sequential updates.
result Existing heuristic formulas can be replicated as ISU decompositions.
The paper defines and proves the existence of decompositions of integral varifolds.
problem Existence of integral varifold decompositions.
method Introducing and proving the existence of decompositions of integral varifolds into countably many integral varifolds.
result Existence of decompositions of integral varifolds whose first variation is representable by integration.
We give an example of two JSJ decompositions of a group that are not related by conjugation, conjugation of edge-inclusions, and slide moves. This answers the question of Rips and Sela stated in "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition," Ann. of Math. 146 (1997), 53-109. On th…
We consider a union of two pants decompositions of the same orientable 2-dimensional surface of any genus g. Each pants decomposition corresponds to some handlebody bounded by this surface, so two pants decompositions correspond to a Heegaard splitting of a 3-manifold. We introduce a groupoid FT acting on double pants …
We present a novel nonnegative tensor decomposition method, called Legendre decomposition, which factorizes an input tensor into a multiplicative combination of parameters. Thanks to the well-developed theory of information geometry, the reconstructed tensor is unique and always minimizes the KL divergence from an inpu…
Handles decompositions reveal new open book structures.
problem Understanding and manipulating open book structures on manifolds.
method Build handle decompositions and describe handle slides.
result Stabilization of open books to trivial monodromy.
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…
Unified algorithm for tensor decomposition supports multiple loss functions and models.
problem Efficient tensor decomposition for various models and loss functions.
method Hierarchical combination of ADMM and MM for optimization.
result Wide-range applications can be solved by the proposed algorithm.
The paper defines and proves stabilization for 3-manifold decompositions with multibranched surface intersections.
problem Decomposing 3-manifolds with more than 3 handlebodies and multibranched surface intersections.
method Definition and proof of stabilization operations for these decompositions.
result Stable equivalence of handlebody decompositions with multibranched surface intersections.
To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary. A new formulation of sparse coding is proposed which enables high dimensional tensors dictiona…
Analyzes canonical reductive decomposition of extrinsic homogeneous submanifolds.
problem Understanding the reductive decomposition of extrinsic homogeneous submanifolds.
method Examines Lie subgroups and reductive decompositions of homogeneous structures.
result Establishes a connection with the Ambrose-Singer theorem and homogeneous structures.
Tensor decomposition methods are widely used for model compression and fast inference in convolutional neural networks (CNNs). Although many decompositions are conceivable, only CP decomposition and a few others have been applied in practice, and no extensive comparisons have been made between available methods. Previo…
Examines respectful decompositions of Lie algebras.
problem Understanding respectful decompositions of Lie algebras.
method Analyzes vector space decomposition properties of Lie algebras.
result Basic properties of respectful decompositions are examined.
In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition…
Formula connects foliated simplicial volume with group cost.
problem Calculating integral foliated simplicial volume.
method Ergodic decomposition formula for simplicial volume.
result Integration formula linking foliated simplicial volume and group cost.
The study examines how different interpolation methods affect the decomposition of life insurance surplus.
problem The impact of different interpolation methods on the decomposition of life insurance surplus.
method The study uses the IASU decomposition method to analyze the effects of different interpolation methods (Lee-Carter and linear) on the surplus decomposition.
result Lee-Carter and linear interpolation yield almost identical decompositions, while constant approximations result in different decompositions.
The paper uses tensor decompositions to improve neural network models for tree data.
problem Encoding structural knowledge from tree-structured data efficiently.
method Introduces new aggregation functions using Canonical and Tensor-Train decompositions.
result Proposed models outperform traditional methods on tree classification tasks.
Analyzes vector fields in polytope decompositions, proving curve finiteness.
problem Analyzing vector fields in polytope decompositions.
method Proves integral curves are chopped into finitely many pieces by polytope decompositions.
result Finiteness of edge flips in discrete Yamabe flow.
New surfaces defy traditional pants decompositions.
problem Creating surfaces without bounded pants decompositions.
method Constructing quasiconformally homogeneous hyperbolic Riemann surfaces.
result Found surfaces with infinite type and compact boundary components that defy pants decompositions.
Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
problem Decomposing hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
method Two different approaches to demonstrate the existence of polyhedral decompositions.
result The number of polyhedral decompositions of M is finite.