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 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.
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.
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.
Often, large, high dimensional datasets collected across multiple modalities can be organized as a higher order tensor. Low-rank tensor decomposition then arises as a powerful and widely used tool to discover simple low dimensional structures underlying such data. However, we currently lack a theoretical understanding …
New method improves tensor completion for weakly-dependent spatiotemporal data.
problem Improving tensor completion for weakly-dependent data on graphs.
method Introducing L1-norm and Graph Laplacian penalties for low-rank tensor decomposition and completion. result Improved performance in metro passenger flow prediction.
Large CNNs have delivered impressive performance in various computer vision applications. But the storage and computation requirements make it problematic for deploying these models on mobile devices. Recently, tensor decompositions have been used for speeding up CNNs. In this paper, we further develop the tensor decom…
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.
Robust tensor CP decomposition involves decomposing a tensor into low rank and sparse components. We propose a novel non-convex iterative algorithm with guaranteed recovery. It alternates between low-rank CP decomposition through gradient ascent (a variant of the tensor power method), and hard thresholding of the resid…
A low-rank tensor model simplifies multi-dimensional Markov chains.
problem Simplifying the dynamics of multi-dimensional Markov chains.
method Low-rank tensor decomposition for multi-dimensional state spaces.
result Our tensor model requires fewer parameters and samples than conventional methods.
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.
We propose a new framework for the analysis of low-rank tensors which lies at the intersection of spectral graph theory and signal processing. As a first step, we present a new graph based low-rank decomposition which approximates the classical low-rank SVD for matrices and multi-linear SVD for tensors. Then, building …
In tensor completion tasks, the traditional low-rank tensor decomposition models suffer from the laborious model selection problem due to their high model sensitivity. In particular, for tensor ring (TR) decomposition, the number of model possibilities grows exponentially with the tensor order, which makes it rather ch…
Unified framework for non-Euclidean CPD under scalable stochastic mirror descent.
problem Handling non-Euclidean losses in tensor decomposition.
method Tensor fiber sampling strategy-based stochastic mirror descent.
result Global convergence to a stationary point under reasonable conditions.
Tensorized random projections reduce high-dimensional tensor size efficiently.
problem Efficiently reducing the dimension of very high-dimensional tensors.
method Proposes two tensorized random projection maps using TT and CP decompositions.
result TT format offers superior performance in terms of required random projection size.
A new framework improves tensor completion accuracy by considering numerical priors.
problem Tensor completion accuracy loss due to ignoring numerical priors.
method Generalized CP Decomposition Tensor Completion (GCDTC) framework incorporating numerical priors.
result GCDTC framework outperforms state-of-the-arts in non-negative tensor completion.
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. Tensor completion estimates missing components by exploiting the low-rank structure of multi-way data. The recently proposed methods based on tensor train (TT) and tensor ring (TR) show better performance in image recovery than classical ones. Compared with TT and TR, the projected entangled pair state (PEPS), which is…
Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to n⌊p/2⌋ for a p-th order tensor in Rnp. Previously no efficient algorithm can decompose 3rd order ten…
New algorithm recovers tensor factors from incomplete measurements efficiently.
problem Recovering tensor factors from incomplete measurements.
method Scaled gradient descent (ScaledGD) algorithm with spectral initializations.
result ScaledGD provably converges linearly for tensor completion and regression.
We consider the problem of online subspace tracking of a partially observed high-dimensional data stream corrupted by noise, where we assume that the data lie in a low-dimensional linear subspace. This problem is cast as an online low-rank tensor completion problem. We propose a novel online tensor subspace tracking al…
We introduce a new parameterization method for deep learning layers using spectral tensor train decomposition.
problem Efficiency and stability in deep learning models with weight matrix compression.
method Spectral Tensor Train Parameterization (STTP) of weight matrices.
result Improved compression and training stability in neural networks.
We investigate the sample size requirement for exact recovery of a high order tensor of low rank from a subset of its entries. In the Tucker decomposition framework, we show that the Riemannian optimization algorithm with initial value obtained from a spectral method can reconstruct a tensor of size $n\times n \times\c…
A new method for decomposing non-negative tensors using energy-based modeling.
problem Challenges in traditional tensor decomposition methods, especially global optimization and rank selection.
method Energy-based modeling of tensors, considering interactions between modes for global optimization.
result Demonstrates effectiveness in tensor completion and approximation, revealing a relationship between many-body and low-rank approximations.
We study the problem of learning a distribution from samples, when the underlying distribution is a mixture of product distributions over discrete domains. This problem is motivated by several practical applications such as crowd-sourcing, recommendation systems, and learning Boolean functions. The existing solutions e…
Tensorized Rademacher projections outperform Gaussian projections in reducing tensor dimensions.
problem Reducing the dimension of high-dimensional tensors for machine learning.
method Tensorized Rademacher random projections using Tensor Train decomposition.
result Tensorized Rademacher projections can replace Gaussian projections in tensor compression.
New method uses tensor decompositions to overcome the curse of dimensionality for large-scale learning.
problem Large-scale machine learning problems with kernel methods.
method Deterministic Fourier features combined with low-rank tensor decomposition for tensor product structure.
result Demonstrated consistent performance and superior results compared to random Fourier features.
ScaledGD algorithm estimates low-rank tensors efficiently from corrupted data.
problem Estimating meaningful information from corrupted tensor data.
method Scaled gradient descent (ScaledGD) algorithm with tailored spectral initializations.
result ScaledGD achieves linear convergence at a constant rate independent of condition number.
Bayesian model improves image completion accuracy by automatically learning low rank structure.
problem Improving image completion accuracy with limited data and avoiding overfitting.
method Developed a Bayesian low rank tensor ring model with multiplicative interaction and Student-T distribution for sparse core factors.
result The proposed method outperforms state-of-the-art image completion techniques, especially in recovery accuracy.
This work improves fair tensor decomposition using a kernel criterion.
problem Learning fair low-rank tensor decompositions with statistical parity.
method Regularizes Canonical Polyadic Decomposition with KHSIC to ensure approximate statistical parity.
result The proposed algorithm achieves better fairness and fit than state-of-the-art FATR.
Low-rank tensor completion recovers missing entries based on different tensor decompositions. Due to its outstanding performance in exploiting some higher-order data structure, low rank tensor ring has been applied in tensor completion. To further deal with its sensitivity to sparse component as it does in tensor princ…
This paper reviews methods for discovering patient subgroups from EHR data.
problem Discovering subgroups of patients and co-occurring medical conditions from EHR data.
method Low-rank data approximation methods like matrix and tensor decompositions.
result These methods provide transparent and interpretable insights into patient phenotypes.
New method for tensor completion from specific mode observations.
problem Recovering multiway data tensors from partial observations.
method Tensor train decomposition for fiber-wise observations.
result Deterministic recovery guarantees for specific observation patterns.
New method selects features via tensor decomposition and submodular optimization.
problem Feature selection for high-dimensional data.
method Low-rank tensor model, submodular optimization, greedy algorithm.
result Proposed method outperforms state-of-the-art feature selection.
We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms,…
The paper proposes a method to estimate tensor regression parameters using low-rank and sparse Tucker decompositions.
problem Estimating tensor regression parameters from limited data.
method Low-rank and sparse Tucker decompositions, non-convex optimization, projected gradient descent.
result The method can linearly converge to an appropriate solution under certain conditions.
Optimizes tensor completion using geodesics on Segre manifolds.
problem Incomplete tensor data in recommender systems and spectroscopy.
method Riemannian conjugate gradient optimization with explicit geodesic expressions.
result Recovery of tensor decomposition from as little as 10% of data.
Method estimates joint probability density from samples using low-rank decomposition and random projections.
problem Estimating joint probability density from limited samples.
method Low-rank tensor decomposition, dictionaries, and Radon transforms.
result Algorithm outperforms previous methods in estimating synthetic probability densities.
Estimates joint probability distribution from 1-way marginals using low-rank tensors and random projections.
problem Nonparametric estimation of joint probability mass function (PMF) from limited data.
method Low-rank tensor decomposition and random projections to link data to PMF estimation.
result Estimates joint density from 1-way marginals using transformed space and novel algorithm.
Paper proposes a novel MTL framework for personalized modeling of diverse individuals.
problem Personalized modeling of heterogeneous subpopulations with high-dimensional data.
method Low-rank tensor decomposition for multi-task learning.
result Superior performance compared to benchmarks in diverse subpopulation scenarios.
Tensors play a central role in many modern machine learning and signal processing applications. In such applications, the target tensor is usually of low rank, i.e., can be expressed as a sum of a small number of rank one tensors. This motivates us to consider the problem of low rank tensor recovery from a class of lin…
TensorGuide improves LoRA efficiency and expressivity through joint tensor-train optimization.
problem Limited expressivity and generalization of standard LoRA.
method TensorGuide uses a unified tensor-train structure with controlled Gaussian noise to generate correlated low-rank matrices.
result TensorGuide achieves superior accuracy and scalability with fewer parameters compared to standard LoRA and TT-LoRA.
Tensor decomposition on big data has attracted significant attention recently. Among the most popular methods is a class of algorithms that leverages compression in order to reduce the size of the tensor and potentially parallelize computations. A fundamental requirement for such methods to work properly is that the lo…
Gradient descent can find better tensor decompositions than lazy training in over-parameterized settings.
problem Finding better tensor decompositions in over-parameterized settings.
method Gradient descent on over-parameterized tensor decomposition problems.
result Gradient descent can find an approximate tensor decomposition with rank m=O∗(r2.5llogd), while lazy training requires m=Ω(dl−1). 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.
This paper introduces a new multivariate convolutional sparse coding based on tensor algebra with a general model enforcing both element-wise sparsity and low-rankness of the activations tensors. By using the CP decomposition, this model achieves a significantly more efficient encoding of the multivariate signal-partic…
Efficient video captioning model captures cross-modal interactions.
problem Capturing frame-level cross-modal interactions in video captioning.
method Proposes High-Order Cross-Modal Attention (HOCA) and Low-Rank HOCA.
result Low-Rank HOCA achieves state-of-the-art performance.
Develops SymGCP for tensor decompositions with general symmetry.
problem Handling symmetry in tensor decompositions for better model accuracy.
method Introduces SymGCP, a generalized CP decomposition that accounts for any subset of tensor modes' symmetry.
result SymGCP enables efficient and scalable tensor decomposition with improved model robustness and accuracy.