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 ℓ0 norm with nonnegativity constraints. result Error bounds and minimax lower bounds are established for the proposed model.
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 ℓ0 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.
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.
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.
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.
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.
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…
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.
Proposes CC-NMDF for analyzing manifold-valued data.
problem Nonlinear structure in manifold-valued data requires new analysis methods.
method Curvature-corrected nonnegative manifold data factorization (CC-NMDF) with an iterative algorithm.
result Demonstrates CC-NMDF on real-world diffusion tensor MRI data.
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.
This paper is concerned with improving the empirical convergence speed of block-coordinate descent algorithms for approximate nonnegative tensor factorization (NTF). We propose an extrapolation strategy in-between block updates, referred to as heuristic extrapolation with restarts (HER). HER significantly accelerates t…
There is currently an unprecedented demand for large-scale temporal data analysis due to the explosive growth of data. Dynamic topic modeling has been widely used in social and data sciences with the goal of learning latent topics that emerge, evolve, and fade over time. Previous work on dynamic topic modeling primaril…
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.
We augment the nonnegative matrix factorization method for audio source separation with cues about directionality of sound propagation. This improves separation quality greatly and removes the need for training data, with only a twofold increase in run time. This is the first method which can exploit directional inform…
It has been recently shown that sparse, nonnegative tensor factorization of multi-modal electronic health record data is a promising approach to high-throughput computational phenotyping. However, such approaches typically do not leverage available domain knowledge while extracting the phenotypes; hence, some of the su…
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.
Introduces nondecreasing rank for matrices and tensors, developing methods and applications.
problem Finding low-rank approximations for matrices and tensors with monotonic constraints.
method Developed a variant of hierarchical alternating least squares algorithm for finding low ND rank approximations.
result Low ND rank factorizations can be found and interpreted for real-world datasets.
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…
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.
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.
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.
Joint analysis of data from multiple information repositories facilitates uncovering the underlying structure in heterogeneous datasets. Single and coupled matrix-tensor factorization (CMTF) has been widely used in this context for imputation-based recommendation from ratings, social network, and other user-item data. …
In this note we classify compact 4-manifolds with harmonic Weyl tensor and nonnegative biorthogonal curvature
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 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
We introduce a dynamic generative model, Bayesian allocation model (BAM), which establishes explicit connections between nonnegative tensor factorization (NTF), graphical models of discrete probability distributions and their Bayesian extensions, and the topic models such as the latent Dirichlet allocation. BAM is base…
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…
Nonnegative matrix factorization (NMF) has been widely used in machine learning and signal processing because of its non-subtractive, part-based property which enhances interpretability. It is often assumed that the latent dimensionality (or the number of components) is given. Despite the large amount of algorithms des…
In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product R×Nn−1, or globally conformally equivalent to the Euclidean space Rn or to the round sphere Sn. In particular, we show that any comple…
In this letter, we propose a new identification criterion that guarantees the recovery of the low-rank latent factors in the nonnegative matrix factorization (NMF) model, under mild conditions. Specifically, using the proposed criterion, it suffices to identify the latent factors if the rows of one factor are \emph{suf…
Graph neural networks speed up nonnegative matrix factorization.
problem Efficiently factorize nonnegative matrices for various applications.
method Developed a graph neural network that combines bipartite self-attention with ADMM updates.
result Significant acceleration achieved in nonnegative matrix factorization.
The exact nonnegative matrix factorization (exact NMF) problem is the following: given an m-by-n nonnegative matrix X and a factorization rank r, find, if possible, an m-by-r nonnegative matrix W and an r-by-n nonnegative matrix H such that X=WH. In this paper, we propose two heuristics for exac…
An algorithm for computing positive semidefinite factorizations of matrices.
problem Computing positive semidefinite factorizations of matrices.
method Non-commutative extension of Lee-Seung's algorithm (Matrix Multiplicative Update, MMU).
result The MMU algorithm ensures PSD updates and achieves critical points.
Bayesian NMF model improves predictions and avoids overfitting.
problem Predicting missing values and finding hidden patterns in nonnegative data.
method Flexible and hierarchical prior for Bayesian NMF with Gibbs sampling.
result The proposed model leads to better predictions and avoids overfitting.
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.
A new NMF variant tackles underdetermined problems with sparse and separable assumptions.
problem Underdetermined blind source separation, especially multispectral image unmixing.
method Sparse Separable Nonnegative Matrix Factorization (SSNMF) combining separability and sparsity assumptions. Algorithm based on SNPA and sparse nonnegative least squares.
result In noiseless settings, the algorithm recovers true underlying sources.
New NMF algorithm uses Toeplitz matrix for facial recognition.
problem Facial recognition performance improvement.
method Proposes TNMF algorithm with Toeplitz penalty for NMF.
result TNMF outperforms ZNMF and other constrained NMF algorithms.
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.
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…
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 f is …
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.
New algorithm completes nonnegative tensors with fewer samples and faster convergence.
problem Tensor completion tension between sample complexity and computational complexity.
method Integer programming and Blended Conditional Gradients algorithm.
result Achieves information-theoretic sample complexity rate with practical convergence.
This paper classifies solitons under specific tensor conditions.
problem Classifying solitons under vanishing conditions on the Weyl, Cotton, and Cao-Chen tensors.
method Analyzing complete conformal gradient solitons and using tensor conditions.
result Classification of complete nontrivial locally conformally flat conformal gradient solitons.
Nonnegative matrix factorization (NMF), a dimensionality reduction and factor analysis method, is a special case in which factor matrices have low-rank nonnegative constraints. Considering the stochastic learning in NMF, we specifically address the multiplicative update (MU) rule, which is the most popular, but which h…
ZNMF improves facial recognition performance using data-dependent penalties.
problem Facial recognition performance in the Cambridge ORL database.
method ZNMF uses data-dependent auxiliary constraints to modify NMF.
result ZNMF outperforms other constrained NMF algorithms in facial recognition.
Nonnegative Boltzmann machines (NNBMs) are recurrent probabilistic neural network models that can describe multi-modal nonnegative data. NNBMs form rectified Gaussian distributions that appear in biological neural network models, positive matrix factorization, nonnegative matrix factorization, and so on. In this paper,…
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.