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…
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48 results for “nonnegative tensor decomposition”
Paper studies nonnegative Tucker decomposition identifiability with sparsity conditions.
problem Identify nonnegative Tucker decomposition factors uniquely.
method Adapting NMF identifiability results, derive procedures using tensor unfoldings or slices.
result Nonnegative Tucker decomposition factors are identifiable under certain sparsity conditions.
Proposes a method for tensor completion with sparse factors and missing data.
problem Recovering nonnegative data from noisy observations with missing values.
method Sparse nonnegative Tucker decomposition with norm for sparsity, maximum likelihood estimation, and error bounds.
result The method outperforms existing tensor-based or matrix-based methods in nonnegative tensor data completion.
Paper proposes ONTD for nonnegative tensor data.
problem Handling nonnegative tensor data efficiently.
method Orthogonal Nonnegative Tucker Decomposition (ONTD) with convex relaxation algorithm.
result Demonstrates effectiveness on real-world image data applications.
The paper improves density estimation in high dimensions using tensor decompositions.
problem Density estimation struggles in high-dimensional data due to the curse of dimensionality.
method The paper uses nonnegative tensor decompositions to simplify dependence assumptions and estimate marginal distributions.
result Theoretical results show that restricting estimation to low-rank nonnegative PARAFAC or Tucker decompositions removes the dimensionality exponent on bin width rates.
New hierarchical tensor decomposition model for complex data.
problem Lack of natural generalization of hierarchical NMF to tensors.
method Proposes a new hierarchical nonnegative tensor decomposition (HNTF) model.
result Model more naturally illuminates topic hierarchy.
Nonnegative CANDECOMP/PARAFAC (NCP) decomposition is an important tool to process nonnegative tensor. Sometimes, additional sparse regularization is needed to extract meaningful nonnegative and sparse components. Thus, an optimization method for NCP that can impose sparsity efficiently is required. In this paper, we co…
New method improves dynamic topic modeling for large-scale data.
problem Lack of temporal information in dynamic topic modeling.
method Nonnegative CP tensor decomposition (NNCPD) for data tensor.
result Significantly improved results compared to NMF-based methods.
Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of data. However, as the data tensor often has multiple modes and is large-scale, exist…
Decomposes submanifolds with special tensors into simpler parts.
problem Understanding the structure of submanifolds with special tensors.
method Established a decomposition theorem for submanifolds with nonnegative sectional curvature and a Codazzi tensor with parallel mean curvature.
result Submanifolds with these tensors are locally isometric to a direct product of irreducible factors.
Method determines latent dimensionality in international trade flows.
problem Finding meaningful low-dimensional latent features in high-dimensional international trade data.
method Proposes a latent dimension determination method based on clustering of nonnegative RESCAL decompositions.
result Validates the latent features against empirical economic facts.
New algorithm learns interpretable CP-basis from streaming tensor data under Markovian constraints.
problem Learning interpretable CP-basis from streaming tensor data under Markovian constraints.
method Online Tensor Factorization (OTF) with CANDECOMP/PARAFAC (CP) decomposition, proving convergence to stationary points.
result Algorithm converges almost surely to stationary points of the objective function under Markovian data generation.
DeepTensor uses deep networks to efficiently decompose tensors with improved performance and robustness.
problem Efficiently decomposing tensors with deep learning to capture nonlinear structures.
method Low-rank tensor decomposition using deep generative networks trained to minimize approximation error.
result DeepTensor outperforms classical methods like SVD and PCA in various applications, including image denoising and 3D MRI.
The paper tackles tensor factorization and completion from noisy data.
problem Sparse nonnegative tensor factorization and completion from partial and noisy observations.
method Minimizes the sum of maximum likelihood estimation and tensor norm with nonnegativity constraints.
result Error bounds and minimax lower bounds are established for the proposed model.
Modeling variability in tensor decomposition methods is one of the challenges of source separation. One possible solution to account for variations from one data set to another, jointly analysed, is to resort to the PARAFAC2 model. However, so far imposing constraints on the mode with variability has not been possible.…
Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.
problem Characterizing gradient ρ-Einstein solitons with specific tensor properties.
method Analyzing the properties of Bach tensor and using local warping to classify solitons.
result Gradient ρ-Einstein solitons with radially nonnegative Bach tensor are locally warped products of an interval and an Einstein manifold.
Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvaturemath.DG
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
New algorithm for nonnegative tensor completion with linear convergence rate.
problem Tensor completion without known optimal sample complexity rate.
method Integer optimization using a specific 0-1 polytope gauge norm.
result Achieves information-theoretic rate with linear convergence.
NNEinFact fits any nonnegative tensor factorization quickly and accurately.
problem Limited user-friendly tools for fitting tailored nonnegative tensor factorizations.
method NNEinFact is an einsum-based multiplicative update algorithm that fits any nonnegative tensor factorization.
result NNEinFact converges to a stationary point of the loss, supports missing data, and fits tensors with hundreds of millions of entries in seconds.
Enhances tensor regression for interpretability and performance.
problem Interpreting and modeling multidimensional tensor data with structural heterogeneity.
method Generalized Nonnegative Structured Kruskal Tensor Regression (NS-KTR) with hybrid regularization and nonnegativity constraints.
result NS-KTR outperforms conventional methods in synthetic and real hyperspectral datasets.
This work proposes a new method to estimate joint probability from pairwise marginals, reducing sample complexity.
problem Direct nonparametric estimation of high-dimensional joint probability is infeasible due to the curse of dimensionality.
method Developed a coupled nonnegative matrix factorization (CNMF) framework using only pairwise marginals.
result The method provably recovers the joint probability mass function up to bounded error in finite iterations under reasonable conditions.
Paper speeds up tensor factorization algorithms.
problem Improving convergence speed of tensor factorization algorithms.
method Proposes an extrapolation strategy between block updates.
result HER significantly accelerates convergence speed.
In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature
We propose a completely unsupervised method to understand audio scenes observed with random microphone arrangements by decomposing the scene into its constituent sources and their relative presence in each microphone. To this end, we formulate a neural network architecture that can be interpreted as a nonnegative tenso…
Proves inequality for tensor fields on curved spaces.
problem Generalizing inequality for tensor fields on curved spaces.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proves Michael-Simon-Sobolev inequality for tensor fields.
Alexandrov spaces with non negative curvature and displacement convexity of the entropy tensormath.DG
Alexandrov spaces with non-negative curvature are characterized by the matrix displacement convexity of an entropy tensor.
problem Characterizing non-negative curvature in Alexandrov spaces
method Constructing a parallel trivialization of the entropy tensor
result The entropy tensor is matrix displacement convex on Alexandrov spaces
Efficiently factorizes coupled matrix tensor data for better accuracy and speed.
problem Poor computation efficiency in existing N-CMTF algorithms.
method Column-wise element selection to prevent frequent gradient updates.
result More accurate and computationally efficient factorization.
Nonnegative matrix factorization (NMF) is a powerful tool for data mining. However, the emergence of `big data' has severely challenged our ability to compute this fundamental decomposition using deterministic algorithms. This paper presents a randomized hierarchical alternating least squares (HALS) algorithm to comput…
Tensor decomposition recovers Gaussian mixtures from moments.
problem Recovering Gaussian mixture models from datasets.
method Symmetric tensor decomposition of moment tensors built from empirical moments.
result Identifiable tensors with interpolation degree less than half their order.
Efficient NTF algorithm for large sparse tensors.
problem Sparse multi-dimensional data and limitations of existing NTF algorithms.
method Saturating Coordinate Descent with element selection based on Lipschitz continuity.
result Proposes a scalable NTF algorithm for large tensors.
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.
This article is motivated by soccer positional passing networks collected across multiple games. We refer to these data as replicated spatial passing networks---to accurately model such data it is necessary to take into account the spatial positions of the passer and receiver for each passing event. This spatial regist…
In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product , or globally conformally equivalent to the Euclidean space or to the round sphere . In particular, we show that any comple…
NACT improves tensor regression predictions with regularization.
problem Improving tensor regression predictions with structural information.
method Noise-Augmented regularization on Tucker decomposition.
result Achieves exact regularization on core tensor in linear and generalized linear tensor regression.
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…
Paper bounds tensor decomposition's RLCT, aiding Bayesian inference.
problem Unclear mathematical property of tensor decomposition.
method Algebraic geometrical method for upper bound derivation.
result Upper bound of real log canonical threshold (RLCT) derived.
Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of…
The paper shows how gradient flow on over-parametrized tensor decomposition behaves like deflation.
problem Understanding the training dynamics of gradient flow on tensor decomposition.
method Empirical observation and mathematical proof of gradient flow dynamics for orthogonally decomposable tensors.
result Gradient flow dynamics for orthogonally decomposable tensors follows a tensor deflation process, recovering all tensor components.
Matrix factorizations and their extensions to tensor factorizations and decompositions have become prominent techniques for linear and multilinear blind source separation (BSS), especially multiway Independent Component Analysis (ICA), NonnegativeMatrix and Tensor Factorization (NMF/NTF), Smooth Component Analysis (Smo…
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.
The paper explores tensor decompositions in deep learning models.
problem Compressing parameter space and creating richer representations.
method Tensor decompositions applied to deep learning models.
result Tensor methods can yield richer adaptive representations of complex data.
Exact nuclear norm, completion and decomposition for random overcomplete tensors via degree-4 SOScs.DS
New algorithms solve tensor problems with random components using SDP.
problem Exact tensor nuclear norm, decomposition, and completion for random tensors.
method Degree-4 Sum of Squares (SOS) semidefinite programs.
result Exact solutions for tensor nuclear norm, decomposition, and completion with random asymmetric components.
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.
The report analyzes Legendre decomposition for tensor data.
problem Finding effective lower dimensional representations of tensors.
method Theoretical analysis of dual parameters and dually flat manifold properties, followed by experimental verification and clustering.
result Parameters on submanifold cannot be directly used as low-rank representations.
Develops a tensor decomposition method with side information.
problem Identifying the relationship between a high-dimensional tensor and side information.
method Supervised tensor decomposition incorporating multiple feature matrices.
result Captures effective dimension reduction of the data tensor in feature space.
The paper classifies compact quasi-Einstein manifolds with boundary.
problem Classifying compact quasi-Einstein manifolds with boundary.
method Analyzing manifolds with nonnegative sectional curvature and zero radial Weyl tensor.
result Classification of quasi-Einstein manifolds, including standard hemisphere and new examples.
In this paper, we propose a new fast and robust recursive algorithm for near-separable nonnegative matrix factorization, a particular nonnegative blind source separation problem. This algorithm, which we refer to as the successive nonnegative projection algorithm (SNPA), is closely related to the popular successive pro…
In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of complete 3-manifolds with nonnegative Bakry-Emery Ricci curvature assuming that is …