Scalable and robust TR decomposition for large-scale data with missing entries and outliers.
problem Handling large-scale tensor data with missing entries and outliers.
method Auto-weighted steepest descent method for missing entries and outliers identification, FGMC and RStS strategies.
result Outperforms existing TR decomposition methods in the presence of outliers and runs faster than robust tensor completion algorithms.
A new tensor ring mixture model improves density estimation efficiency.
problem Efficient probability density estimation in statistical machine learning.
method Tensor ring decomposition with mixture model for adaptive weights.
result Enhanced expressive capability and flexibility in density estimation.
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…
BRTR improves robust tensor completion with automatic rank detection.
problem Robust tensor completion from incomplete data with outliers.
method Bayesian robust tensor ring decomposition (BRTR) with variational Bayesian (VB) algorithm.
result Automatic detection of TR rank and improved performance over state-of-the-art methods.
Unified framework for coupled tensor completion improves recovery accuracy.
problem Improving recovery accuracy in coupled tensor completion.
method Unified framework using tensor ring (TR) decomposition with shared latent factors and novel optimization model.
result The proposed method achieves superior recovery accuracy on real-world data compared to state-of-the-art methods.
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.
Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S_r] of a symmetric group S_r. If for a class of tensors T such a W is known, the elements of the orthogonal subspace W^{\bot} of W within the dual space of K[S_r] yield linear identities needed for a t…
Tensor completion recovers a multi-dimensional array from a limited number of measurements. Using the recently proposed tensor ring (TR) decomposition, in this paper we show that a d-order tensor of dimensional size n and TR rank r can be exactly recovered with high probability by solving a convex optimization program,…
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…
Bayesian Tensor Ring factorization improved for scalability and handling of discrete data.
problem Scalability issues and handling of discrete data in Bayesian Tensor Ring factorization.
method Proposes a novel Bayesian Tensor Ring model with a nonparametric Multiplicative Gamma Process prior and Pólya-Gamma augmentation for discrete data. Developed efficient Gibbs sampler and online EM algorithm for scalability.
result Significantly improved scalability and handling of discrete data compared to previous methods.
Paper proposes a new model for noisy tensor completion.
problem Handling noise in tensor completion.
method Tensor ring nuclear norm (TRNN) and least-squares estimator.
result Effective recovery of noisy incomplete tensor data.
Adaptive algorithm learns tensor network structures from data.
problem Identifying optimal tensor network structure from data.
method Greedy approach starting from rank one tensor, small rank increments.
result Adaptive algorithm identifies efficient tensor network structures.
Algorithm learns polynomial transformations of Gaussian distributions.
problem Learning high-dimensional polynomial transformations of Gaussian distributions.
method Polynomial-time algorithms for smoothed settings, tensor ring decomposition.
result First end-to-end guarantees for learning pushforwards under neural networks.
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…
A new method uses CPD to efficiently model feature interactions in non-sequential data.
problem Efficiently modeling feature interactions in non-sequential data with high computational and memory costs.
method Implicitly represent model parameters as a tensor, factorize into a compact Tensor Train (TT) format, and use Canonical Polyadic (CP) Decomposition for invariance to feature ordering.
result The proposed CP-based predictor outperforms other TN-based predictors on sparse data and matches neural network performance on dense non-sequential tasks.
T-Basis represents neural network tensors with fewer parameters.
problem Efficiently representing neural network tensors with fewer parameters.
method T-Basis uses Tensor Rings to represent tensors in a neural network, parameterizing them with a small number of coefficients.
result T-Basis achieves high compression rates with minimal performance loss.
We show that near-horizon geometries in the presence of a positive cosmological constant cannot exist with ring topology. In particular, de Sitter black rings with vanishing surface gravity do not exist. Our result relies on a known mathematical theorem which is a straightforward consequence of a type of energy conditi…
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…
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.
The paper studies the metric and algebraic structures on section rings of projective manifolds.
problem Understanding the relationship between metric and algebraic structures on section rings.
method Analyzes the section ring of projective manifolds and ample line bundles, proving approximate isometry properties under various norms.
result Characterizes L2-norms associated with continuous plurisubharmonic metrics and refines the theorem of Phong-Sturm. 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.
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 finds canonical triangulations for specific 3-manifolds.
problem Finding canonical decompositions for cusped hyperbolic 3-manifolds.
method Showed local convexity at every face of the geometric triangulation.
result Found canonical triangulations for Dehn fillings of the Borromean rings link complement and related manifolds.
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. 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.
In this paper we study the tensor powers of the standard representation of the quantum super-algebra Uq(sl(2∣1), focusing on the rings of its algebra endomorphisms, called centraliser algebras and denoted by LGn. Their dimensions were conjectured by I. Marin and E. Wagner \cite{MW}. We prove this conjecture, desc…
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.
Deep neural networks have demonstrated state-of-the-art performance in a variety of real-world applications. In order to obtain performance gains, these networks have grown larger and deeper, containing millions or even billions of parameters and over a thousand layers. The trade-off is that these large architectures r…
MARS automatically selects tensor decomposition ranks, improving performance in neural network tasks.
problem Determining optimal decomposition ranks in tensor decompositions.
method MARS uses binary masks to learn optimal tensor structure during training via relaxed MAP estimation.
result MARS achieves better results than previous methods in various tasks.
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.
We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.
Two methods preserve tensor structure for reduced dimensionality in tensor regression.
problem Reducing dimensionality of tensor predictors for improved interpretation and accuracy.
method Developed two tensor dimension reduction methods using Tucker and CP decompositions.
result Substantial improvement in accuracy over existing methods in simulations and applications.
Tensor decomposition is an important technique for capturing the high-order interactions among multiway data. Multi-linear tensor composition methods, such as the Tucker decomposition and the CANDECOMP/PARAFAC (CP), assume that the complex interactions among objects are multi-linear, and are thus insufficient to repres…
The study connects norms and filtrations on section rings of projective manifolds.
problem Understanding norms and filtrations on section rings of polarized projective manifolds.
method Analyzes submultiplicative norms and their equivalence to sup-norms, discusses applications to spectral theory and holomorphic extension.
result Injective and projective tensor norms on symmetric algebras are asymptotically equivalent.
The paper studies the geometry of Nakajima quiver varieties and their decompositions.
problem Understanding the geometry of Nakajima quiver varieties and their decompositions.
method Investigates the Białynicki--Birula decomposition of Nakajima quiver varieties, describing fixed points in terms of representations with relations of auxiliary quivers.
result Computes the motivic decomposition of Nakajima quiver varieties in terms of quiver-chain moduli spaces.
Modeling inverse dynamics is crucial for accurate feedforward robot control. The model computes the necessary joint torques, to perform a desired movement. The highly non-linear inverse function of the dynamical system can be approximated using regression techniques. We propose as regression method a tensor decompositi…
Tensors are multidimensional arrays of numerical values and therefore generalize matrices to multiple dimensions. While tensors first emerged in the psychometrics community in the 20th century, they have since then spread to numerous other disciplines, including machine learning. Tensors and their decomposi…
Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
Anomaly Detection has several important applications. In this paper, our focus is on detecting anomalies in seller-reviewer data using tensor decomposition. While tensor-decomposition is mostly unsupervised, we formulate Bayesian semi-supervised tensor decomposition to take advantage of sparse labeled data. In addition…
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.