FunBaT extends Tucker decomposition to handle continuous-indexed tensor data.
problem Handling continuous-indexed tensor data that doesn't fit traditional Tucker decomposition.
method FunBaT treats continuous-indexed data as interactions between a core tensor and a group of latent functions modeled by Gaussian processes (GP). It converts each GP into a state-space prior and uses advanced message-passing techniques for scalable inference.
result FunBaT effectively handles real-world data with continuous indexes, demonstrating its advantage in synthetic and real-world applications.
SDIFT generates full-field dynamics from sparse, irregular data.
problem Modeling and reconstructing physical dynamics from sparse, off-grid observations.
method SDIFT uses a functional Tucker model and sequential diffusion for generating full-field evolution from irregular sparse observations.
result Significant improvements in reconstruction accuracy and computational efficiency compared to state-of-the-art approaches.
Adaptive tensor modeling preserves continuity in multidimensional data.
problem Discretization of continuous multidimensional data loses important information.
method Functional Tucker decomposition (FTD) with RKHS modeling.
result FTD enables adaptive and expressive tensor modeling.
Unified framework for statistical inference of low-rank tensors.
problem Statistical inference for tensors in high-dimensional data.
method Unified framework using debiasing and tangent space projection.
result Achieves asymptotic normality and minimax-optimal confidence intervals.
A new diffusion model generates structured tensors for high-dimensional data.
problem Generating a structured tensor with a target distribution.
method Tucker diffusion model with Tucker-Unet architecture.
result Generated tensors converge to the true data distribution at a rate dependent on tensor mode dimensions.
Extends geometrical description of tensor manifolds in tree-based formats.
problem Geometrical description of tensor manifolds in tree-based formats.
method Provided a new geometrical description of manifolds of tensors in tree-based format.
result Geometrical description compatible with Tucker format.
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.
Knowledge graphs are structured representations of real world facts. However, they typically contain only a small subset of all possible facts. Link prediction is a task of inferring missing facts based on existing ones. We propose TuckER, a relatively straightforward but powerful linear model based on Tucker decomposi…
HOTCAKE compresses CNNs by decomposing kernels into smaller parts.
problem Compressing deep CNNs without significant accuracy loss.
method Input channel decomposition, guided Tucker rank selection, higher order Tucker decomposition, fine-tuning.
result HOTCAKE produces highly compressed CNN models with good accuracy.
In this paper, we study the nonnegative tensor data and propose an orthogonal nonnegative Tucker decomposition (ONTD). We discuss some properties of ONTD and develop a convex relaxation algorithm of the augmented Lagrangian function to solve the optimization problem. The convergence of the algorithm is given. We employ…
Geometrically, tensors of fixed rank form a minimal submanifold.
problem Understanding the geometric properties of tensors of fixed rank.
method Geometric analysis of tensors in Euclidean space.
result Real tensors of fixed multilinear rank form a minimal submanifold.
Estimates low-rank distributional matrices from incomplete samples.
problem Matrix completion for distributional entries with limited observed data.
method Kernel mean embeddings, Tucker rank, functional unfolding operators.
result Effective estimator for distributional matrix completion established.
A new tensor completion method using tensor networks with Tucker wrapper.
problem Low-rank tensor completion in various applications.
method Solving LRTC as a system of nonlinear equations using a two-level alternative least squares method.
result The method converges to the exact solution at a linear rate with high probability.
NA0CT2 improves tensor regression predictions with ℓ0 regularization.
problem Improving tensor regression predictions with structural information.
method Noise-Augmented ℓ0 regularization on Tucker decomposition. result Achieves exact ℓ0 regularization on core tensor in linear and generalized linear tensor regression. New model for network analysis using functional data.
problem Existing network models treat nodes as functions, but this paper introduces functional edges.
method Transform adjacency matrix into functional adjacency tensor, apply Tucker decomposition, regularize basis matrices, and solve tensor completion problem.
result The model effectively captures community structure and handles irregular functional edge data.
SG-NTF completes HDI tensors with spectral mapping and spatio-temporal gating.
problem High-dimensional and incomplete tensor completion.
method Spectra-Guided Neural Tucker Factorization (SG-NTF) with Spatio-Temporal Co-Gating (STCG).
result Maintains competitive completion accuracy with parameter efficiency.
A new kernel improves tensor classification accuracy and reduces computation time.
problem Challenges in classifying high-dimensional tensor data.
method Proposes a weighted subspace exponential kernel based on Tucker decomposition.
result The new kernel outperforms existing methods in accuracy and computational efficiency.
A new probabilistic BTD method for tensor data.
problem Modeling higher-order tensors with robust inference.
method Probabilistic Block-Term Decomposition using variational Bayesian inference and von-Mises Fisher distribution.
result The proposed pBTD can quantify multi-linear structures robustly.
Sparse Tucker decomposition with graph regularization improves time series forecasting accuracy.
problem High-dimensional time series forecasting with over-parameterization issue.
method Sparse Tucker decomposition and graph regularization for tensor-based model.
result Non-asymptotic error bound and superior performance in numerical experiments.
Paper proposes BTuD for unsupervised feature selection.
problem Feature selection in unsupervised learning.
method Bayesian Tucker decomposition (BTuD) with Gaussian residual.
result Successfully applied to various datasets.
Unified model for tensor completion using low-rank and sparse Tucker decomposition.
problem Estimating missing data from incomplete tensor measurements.
method Unified low-rank and sparse enhanced Tucker decomposition model with ADMM.
result Our model achieves higher recovery accuracy on various real-world data sets.
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…
Paper learns meaningful state and action representations from MDP trajectories.
problem Learning good state and action representations from MDP trajectories.
method Tensor decomposition, kernelization, importance sampling, low-Tucker-rank approximation.
result The learned state/action abstractions provide accurate approximations to latent block structures.
Infinite Tucker Decomposition (InfTucker) and random function prior models, as nonparametric Bayesian models on infinite exchangeable arrays, are more powerful models than widely-used multilinear factorization methods including Tucker and PARAFAC decomposition, (partly) due to their capability of modeling nonlinear rel…
Paper proves k-means clustering works on persistence diagrams.
problem Complex geometry of persistence diagram space.
method Proves convergence of k-means on persistence diagram space. result Performance of k-means on persistence diagrams and measures is superior. Paper introduces TSSDMN for modeling dynamic multilayer networks.
problem Capturing temporal and cross-layer dynamics in multilayer networks.
method Tensor State Space Model (TSSDMN) using symmetric Tucker decomposition.
result TSSDMN uniquely captures temporal dynamics within and across layers.
Tucker decomposition is the cornerstone of modern machine learning on tensorial data analysis, which have attracted considerable attention for multiway feature extraction, compressive sensing, and tensor completion. The most challenging problem is related to determination of model complexity (i.e., multilinear rank), e…
Proposes BHT-ARIMA for forecasting multiple short time series.
problem Forecasting multiple short time series with mutual correlations.
method Block Hankel tensors, Tucker decomposition, generalized tensor ARIMA.
result Improves forecasting accuracy and reduces computational cost.
Optimizes neural network training by dynamically updating Tucker decomposition ranks.
problem Redundant parameters in neural network architectures.
method Geometry-aware training of factorized layers in tensor Tucker format.
result Optimal locally approximating the original dynamics without initial rank knowledge.
A new method for traffic data imputation considering spatiotemporal correlations.
problem Traffic data imputation, especially for high-level missing scenarios.
method Spatiotemporal regularized Tucker decomposition approach.
result The proposed method outperforms existing methods on real-world traffic datasets.
This work improves tensor decomposition methods, especially for large datasets.
problem Lack of efficient methods for estimating Tucker decompositions.
method Applies Johnson-Lindenstrauss type guarantees to Tucker decompositions with random embeddings.
result Effective dimension reduction with minimal error for large tensors.
Paper introduces a new histogram estimator for nonparametric density estimation that improves performance.
problem Smoothness-based nonparametric density estimators are not optimal for all types of data.
method Incorporates a multi-view latent variable model into histogram-style estimators.
result A new histogram estimator converges faster to multi-view models in L1 error. Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
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.
Optimization with inequality constraints using embedded gradient vector field method
problem Optimization with inequality constraints
method Geometric framework using quadratic slack variables
result Derives Lagrange multiplier functions and second-order optimality conditions
Gradient descent promotes low-rank solutions in tensor completion.
problem Implicit regularization in tensor factorization using gradient descent.
method Introduced deep Tucker and TensorTrain (TT) unconstrained factorization to address tensor completion.
result Gradient descent promotes solutions with low-rank.
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
In this paper we study a continuous time, optimal stochastic investment problem under limited resources in a market with N firms. The investment processes are subject to a time-dependent stochastic constraint. Rather than using a dynamic programming approach, we exploit the concavity of the profit functional to derive …
We analyze large, multi-dimensional, sparse counting data sets, finding unsupervised groups to provide unique insights into genetic data. We create gene and biological pathway groups based on patients' variants to find common risk factors for four common types of cancer (breast, lung, prostate, and colorectal) and auti…
Proposes a low-rank bilinear pooling model for link prediction in knowledge graphs.
problem Link prediction in incomplete knowledge graphs.
method Factorized bilinear pooling model with Tucker decomposition constraints.
result Efficient and parameter-efficient model with low-rank approximation.
The paper studies neural networks' convergence near origin and saddle points.
problem Directional convergence of neural networks near small initializations and saddle points.
method Gradient flow dynamics analysis of two-homogeneous neural networks.
result Neural networks' weights approximately converge in direction to KKT points for small initializations.
Study KKT conditions for multi-objective optimization on Hadamard manifolds.
problem Optimizing multi-objective interval-valued functions on Hadamard manifolds.
method Developed KKT conditions for Pareto optimal solutions under different ordering and convexity notions.
result Results are more general than on Euclidean spaces.
The paper tackles partial inference in structured prediction using a convex optimization approach.
problem Maximizing a score function with unary and pairwise potentials in graph label spaces.
method Generative model approach with two-stage convex optimization for label recovery.
result Conditions for recovering a majority of labels with provable guarantees.
This paper studies a tensor-structured linear regression model with a scalar response variable and tensor-structured predictors, such that the regression parameters form a tensor of order d (i.e., a d-fold multiway array) in Rn1×n2×⋯×nd. It focuses on the task of estimatin…
We introduce Bayesian Poisson Tucker decomposition (BPTD) for modeling country--country interaction event data. These data consist of interaction events of the form "country i took action a toward country j at time t." BPTD discovers overlapping country--community memberships, including the number of latent com…
ALℓ0CORE tensor decomposition reduces computational cost for sparse count data.
problem Efficiently decompose sparse count data matrices.
method Probabilistic Tucker decomposition with ℓ0-norm constraint. result ALℓ0CORE achieves similar results to full Tucker decomposition at a fraction of the cost. New tensor model reduces GLM estimation error and sample complexity.
problem Estimating GLM coefficients with reduced sample complexity.
method Developed LSR tensor model and block coordinate descent algorithm.
result Minimax lower bound on estimation error, suggesting lower sample complexity.
Paper compresses RNNs using HT decomposition for better performance.
problem Large model sizes of RNNs in sequence analysis.
method Hierarchical Tucker (HT) tensor decomposition for model compression.
result HT-LSTM achieves better compression and accuracy than state-of-the-art methods.