We propose a set of convex low rank inducing norms for a coupled matrices and tensors (hereafter coupled tensors), which shares information between matrices and tensors through common modes. More specifically, we propose a mixture of the overlapped trace norm and the latent norms with the matrix trace norm, and then, w…
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48 results for “matrix norm couplings”
New method for probabilistic clustering using matrix norm couplings.
problem Probabilistic clustering of discrete random variables.
method Local information theoretic approach with convex maximization problem solved via gradient ascent and alternating maximization.
result Our approach is competitive with existing techniques and worthy of further investigation.
A Flexible Optimization Framework for Regularized Matrix-Tensor Factorizations with Linear Couplingscs.LG
Flexible framework for CMTF with ADMM for various constraints and couplings.
problem Challenges in data fusion from multiple sources with varying characteristics.
method Flexible algorithmic framework using AO and ADMM for various constraints, loss functions, and couplings.
result Accurate and computationally efficient results for various loss functions, including KL divergence.
The paper introduces a new method for tail bounds of random vectors and matrices.
problem Estimating norms of random vectors and matrices under moment assumptions.
method Variational tail bounds for norms of random vectors and matrices.
result Dimension-free concentration inequalities for various norms of random vectors and matrices.
Efficiently learns matching rewards in two-sided markets with matrix completion.
problem Learning high-dimensional matching rewards in matching markets with limited data.
method Utilizes matrix completion with a novel approach to handle matching interference.
result Near-optimal guarantees for reward learning under matching interference.
A new method classifies color images using quaternion algebra.
problem Classifying color images with preserved intrinsic relationships.
method LSQMM model with quaternion nuclear norm regularization and ADMM algorithm.
result LSQMM outperforms state-of-the-art methods in classification accuracy and efficiency.
In this paper we introduce and analyze the learning scenario of \emph{coupled nonlinear dimensionality reduction}, which combines two major steps of machine learning pipeline: projection onto a manifold and subsequent supervised learning. First, we present new generalization bounds for this scenario and, second, we int…
Solves steering problem with continuous time, Hilbert-Schmidt cost, and matrix ODEs.
problem Fixed horizon linear quadratic covariance steering in continuous time with a specific terminal cost.
method Formulates necessary conditions as a coupled matrix ODE two-point boundary value problem, designs a matricial recursive algorithm, and proves convergence.
result Proposes and proves the convergence of a matricial recursive algorithm for solving the steering problem.
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.
Regularization of minimization for dealing with outliers and noise in Statistics and Signal Recoverymath.OC
We study the robustness properties of norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the estimator for measurements errors consisting of outliers coupled with noise. We…
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.
Paper proposes an algorithm for PARAFAC2-based CMTF models with various constraints.
problem Jointly analyze matrices and tensors with irregular/ragged data.
method Alternating Optimization (AO) and ADMM for fitting PARAFAC2-based CMTF models with various constraints.
result Accurately recovers underlying patterns using various constraints and linear couplings.
Researchers developed a new Riemannian manifold for SPD matrix-valued optimal transport problems.
problem Optimal transport between SPD matrix-valued measures.
method Formulated as a generalized optimal transport problem with block SPD matrices, endowed with a novel Riemannian manifold structure.
result The novel Riemannian manifold allows solving SPD matrix-valued optimal transport problems using Riemannian optimization.
We introduce a new family of matrix norms, the "local max" norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this…
The Group-Lasso is a well-known tool for joint regularization in machine learning methods. While the l_{1,2} and the l_{1,\infty} version have been studied in detail and efficient algorithms exist, there are still open questions regarding other l_{1,p} variants. We characterize conditions for solutions of the l_{1,p} G…
Incorporates matrix exponential into generative flows for improved performance.
problem Improving generative flow models for better density estimation.
method Integrates matrix exponential into generative flows, proposing new layers and modifying network architecture.
result The proposed model achieves great performance on density estimation.
U-Net trained to recover acoustic interference striations from distorted data.
problem Recovering acoustic interference striations from distorted signals.
method Training a U-Net using a random mode-coupling matrix model to generate training data.
result U-Net successfully recovers AISs under various conditions.
Recently, matrix norm has been widely applied to many areas such as computer vision, pattern recognition, biological study and etc. As an extension of vector norm, the mixed matrix norm is often used to find jointly sparse solutions. Moreover, an efficient iterative algorithm has been designed…
New algorithms estimate matrix norms without matrix multiplication.
problem Estimating matrix norms efficiently in a matrix-free setting.
method Randomized algorithms based on Hutchinson's estimator modifications.
result Oracle complexity bounds for two-to-infinity and one-to-two norms.
New proof shows norms can't explain deep learning's implicit regularization.
problem Understanding the implicit regularization in deep learning.
method Mathematical proof on matrix factorization problems.
result Implicit regularization drives norms towards infinity, suggesting rank minimization is key.
New nonconvex regularizers improve low-rank matrix recovery efficiency and accuracy.
problem Efficiently recover low-rank matrices from incomplete data.
method Factor group-sparse regularization, related to Schatten-p norms.
result Improved generalization error bounds for Schatten-p norms as p decreases.
Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is…
The -support norm is a regularizer which has been successfully applied to sparse vector prediction problems. We show that it belongs to a general class of norms which can be formulated as a parameterized infimum over quadratics. We further extend the -support norm to matrices, and we observe that it is a special …
Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly …
Paper finds a fast method for a matrix norm proximal operator.
problem Optimizing mixed matrix norms efficiently.
method Closed-form computation using soft-thresholding, iterative algorithm for thresholds.
result Mixed prox can be computed in closed form.
We consider the problem of approximately reconstructing a partially-observed, approximately low-rank matrix. This problem has received much attention lately, mostly using the trace-norm as a surrogate to the rank. Here we study low-rank matrix reconstruction using both the trace-norm, as well as the less-studied max-no…
The Schatten quasi-norm can be used to bridge the gap between the nuclear norm and rank function, and is the tighter approximation to matrix rank. However, most existing Schatten quasi-norm minimization (SQNM) algorithms, as well as for nuclear norm minimization, are too slow or even impractical for large-scale problem…
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
The paper tackles matrix estimation from noisy data, focusing on low-rank matrices.
problem Estimating a low-rank matrix from noisy observations.
method The paper analyzes several estimators, including constrained nuclear-norm minimization, nuclear-norm regularized least squares, and a nonconvex constrained low-rank optimization problem.
result The estimators provide upper error bounds that depend on matrix rank, observed fraction, and matrix sums, and are minimax optimal.
The paper analyzes error bounds and KL properties for noisy matrix recovery problems.
problem Noisy low-rank matrix recovery problems.
method Squared F-norm regularization, accelerated alternating minimization method.
result Established error bounds and KL properties for critical points and global minimizers.
Paper tackles low-rank matrix recovery with column -norm regularization.
problem Low-rank matrix recovery problems with column sparsity constraints.
method Developed alternating majorization-minimization (AMM) methods with extrapolation and hybrid AMM.
result Global convergence analysis and superior performance in matrix completion problems.
New coupling matrix manifold improves optimal transport solutions.
problem Optimal transport problems.
method Developed a coupling matrix manifold (CMM) with Riemannian geometry and optimization algorithms.
result Optimization algorithms based on the proposed method perform comparably to classic algorithms and outperform others.
This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery using a random subset of entries observed with additive noise under general non-unif…
SpINNEr uses matrix regression to analyze brain connectivity, improving accuracy over other methods.
problem Analyzing multi-dimensional data like brain imaging arrays using traditional scalar regression methods.
method SpINNEr applies matrix regression with nuclear norm and lasso norms to encourage low rank and sparse solutions.
result SpINNEr outperforms other methods in estimating brain connectivity, especially in well-connected regions.
Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear norm minimization will exactly recover the low rank matrix. In this article, we…
Paper proposes C-STM for multimodal neuroimaging data classification.
problem Multimodal neuroimaging data fusion for better classification.
method Coupled Support Tensor Machine (C-STM) using latent factors from ACMTF.
result C-STM achieves better classification performance than single-mode classifiers.
Linear-cost unbiased estimates for complex models via couplings.
problem High-dimensional Bayesian models with crossed effects and matrix factorization.
method Coupled Gibbs samplers for linear computational cost.
result Unbiased posterior estimates at linear cost.
factor analysismath.OC
New method for factor analysis using nuclear and norms.
problem Finding a low-rank plus sparse decomposition from noisy covariance matrix.
method Formulated an optimization problem with nuclear norm, norm, and KL divergence. Used alternating minimization algorithm.
result Algorithm effectively decomposes covariance matrices in synthetic and real datasets.
Social trust prediction addresses the significant problem of exploring interactions among users in social networks. Naturally, this problem can be formulated in the matrix completion framework, with each entry indicating the trustness or distrustness. However, there are two challenges for the social trust problem: 1) t…
Selective sampling improves matrix completion with known structure.
problem Reconstructing a low-rank matrix with incomplete data.
method Designing observation sets based on matrix structure and selective sampling.
result Improved reconstruction accuracy with selective sampling.
Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing and medical imaging, and this kind of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being non-convex and discont…
Gravitational interactions of higher spin fields are generically plagued by inconsistencies. We present a simple framework that couples higher spins to a broad class of gravitational backgrounds (including Ricci flat and Einstein) consistently at the classical level. The model is the simplest example of a Yang--Mills d…
The spectral -support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral -support norm, whose additional para…
New nonconvex regularizer speeds up low-rank matrix completion.
problem Low-rank matrix completion with good theoretical and empirical performance.
method Proposes a new nonconvex regularizer with adaptive shrinkage, scalable, and fast optimization.
result Proposed method achieves state-of-the-art recovery performance and is the fastest.
Paper proposes DP-Thresholding for estimating sparse high-dimensional covariance matrices with differential privacy.
problem Estimating sparse high-dimensional covariance matrices under differential privacy constraints.
method DP-Thresholding method for achieving non-trivial error bounds.
result DP-Thresholding achieves significant error bounds compared to existing methods.
Numerous applications in data mining and machine learning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a general framework for handling this kind of problems. Nuclear norm based convex surrogate of the rank function in RPCA is widely investigated. Under certain assumptions,…
A new framework uses matrix flows to unify frequentist and Bayesian approaches for sparse GGMs.
problem Challenges in studying conditional independence among many variables with few observations.
method General framework for variational inference with matrix-variate Normalizing Flow in Gaussian Graphical Models.
result Unified benefits of frequentist and Bayesian frameworks for sparse GGMs.
Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalizationstat.ML
Near-interpolating models grow norms quickly, affecting generalization.
problem Understanding the trade-off between interpolation and generalization in near-interpolating models.
method Random matrix theory and eigendecay analysis of data covariance matrix.
result Near-interpolating models exhibit rapid norm growth and worse generalization trade-offs.