Study finds Deep Taylor Decomposition is unreliable for explaining neural networks.
problem Reliability of Deep Taylor Decomposition for explaining neural networks.
method Investigated the theoretical foundations of Deep Taylor Decomposition (DTD) and found it under-constrained.
result DTD is unreliable because its theoretical foundations are under-constrained and roots do not align with input.
A new method explains anomalies in one-class models using deep Taylor decomposition.
problem Understanding why one-class models classify data points as anomalies.
method Recompose one-class SVM as a neural network, perform deep Taylor decomposition.
result The method reliably explains a wide set of data anomalies and outperforms baselines.
Analyzes explaining nonlinear model predictions.
problem Understanding the contribution of inputs to outputs in nonlinear models.
method Merges integrated gradient and deep Taylor decomposition methods.
result Provides a natural reference point for model at use.
Study shows how noise and distractions affect neural network explanations.
problem Lack of consideration for noise and distractions in neural network explanations.
method Examined the impact of noise and distracting elements on neural network interpretation models.
result Noise and distracting elements influence the results of neural network explanations.
A new method exposes motion-related relevance in video frames.
problem Deconstructing relevance in spatio-temporal models for video processing.
method Proposes a discriminative method to separate spatial and temporal relevance.
result Demonstrates effectiveness on UCF-101 action recognition dataset.
Nonlinear methods such as Deep Neural Networks (DNNs) are the gold standard for various challenging machine learning problems, e.g., image classification, natural language processing or human action recognition. Although these methods perform impressively well, they have a significant disadvantage, the lack of transpar…
Deep networks analyze video snippets to predict outcomes, revealing a border effect that can be adjusted for better accuracy.
problem Improving the accuracy of deep networks trained on small video snippets.
method Applied the deep Taylor / LRP technique to understand and identify a border effect, tuning the step size to improve accuracy.
result The step size used to build video snippets can be adjusted to improve deep network accuracy without retraining.
Modified BP attribution methods often ignore later layers' information, leading to misleading explanations.
problem Misleading explanations from modified BP methods ignoring later layers' information.
method Analysis of 9 modified BP methods including Deep Taylor Decomposition, LRP, Excitation BP, PatternAttribution, DeepLIFT, Deconv, RectGrad, Guided BP.
result Only DeepLIFT does not ignore later layers' information, providing a faithful explanation.
Detects and explains anomalies in non-i.i.d data with latent-class dependencies.
problem Detecting and explaining anomalies in data with latent-class dependencies.
method Derives SVDD method to handle latent-class dependency structure, provides probabilistic interpretation.
result Demonstrates effectiveness on real-world offshore data.
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.
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 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.
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.
A new tensor decomposition method using a dictionary for better interpretability.
problem Ensuring interpretability in tensor decomposition models.
method Dictionary-based tensor canonical polyadic decomposition with sparse coding.
result Improves parameter identifiability and estimation accuracy in tensor decomposition.
A new tensor decomposition method that minimizes KL divergence.
problem Tensor reconstruction accuracy.
method Legendre decomposition, based on information geometry.
result Minimizes KL divergence and improves tensor reconstruction accuracy.
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.
Characterizes conditions for quotient spaces of decompositions to be manifolds.
problem Conditions for quotient spaces of decompositions to be manifolds.
method Generalized characterizations of upper semi-continuity for decomposition into one for a class decomposition.
result Characterizations of necessary and sufficient conditions for quotient spaces of decompositions to be k-manifolds (k=1,2). 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.
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…
New tensor network decompositions improve CNN performance.
problem Limited exploration of tensor network decompositions for CNNs.
method Characterized a new class of CNN modules and experimentally compared various decompositions.
result Some nonlinear decompositions outperform existing ones in terms of accuracy and efficiency.
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.
Profinite group analysis reveals manifold decompositions.
problem Decomposing 3-manifolds into simpler parts.
method Profinite completion of fundamental groups.
result Profinite completion determines manifold decompositions.
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.
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.
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.
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.
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). 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 …
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.
Lefschetz decompositions for Kähler manifold eigenforms identified.
problem Understanding the spectrum and eigenspaces of Laplacians on Kähler manifolds.
method Analyzing the eigenspaces of the Laplacian Δk on k-forms on a compact Kähler manifold. result The positive part of the spectrum of Δk lies in the spectrum of Δk+1. 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.
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.
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…
Unique decompositions of 3D manifolds fail in higher dimensions.
problem Uniqueness of decompositions in higher-dimensional manifolds.
method Analyzing decompositions in various dimensions.
result Uniqueness of decompositions fails in higher dimensions.
Hodge decomposition extended to H1 space on non-compact manifolds.
problem Extending Hodge decomposition to non-compact manifolds and Sobolev spaces.
method Generalization of Hodge decomposition to H1 space on non-compact manifolds of nonpositive curvature. result Hodge decomposition established for H1 space on non-compact manifolds of nonpositive curvature. 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.