We theoretically and experimentally investigate tensor-based regression and classification. Our focus is regularization with various tensor norms, including the overlapped trace norm, the latent trace norm, and the scaled latent trace norm. We first give dual optimization methods using the alternating direction method …
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48 results for “latent trace norm”
Proposes a new tensor completion method using dual framework and Riemannian optimization.
problem Low-rank tensor completion with sparse or non-sparse tensor combinations.
method Dual framework, latent trace norm, Riemannian optimization, trust region algorithm.
result Shows the optimal solution lies on a Cartesian product of Riemannian manifolds.
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…
Proposes GTTN for discovering all low-rank structures in deep multi-task learning.
problem Discovering all low-rank structures among tasks in deep multi-task models.
method Introduces GTTN, a convex combination of matrix trace norms of all tensor flattenings, to automatically determine the importance of components.
result Demonstrates the effectiveness of GTTN on real-world datasets.
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…
Proposes a new method for hyperspectral image dimensionality reduction.
problem Highly correlated noisy hyperspectral images.
method Trace Lasso-L1 Graph Cut method using L1-norm for robustness and sparsity.
result Optimal projection matrix maximizing between-class dispersion to within-class dispersion.
This work improves trace norm regularization for multi-task learning with limited data.
problem Learning from few samples across multiple tasks.
method Trace norm regularization for a linear shared representation model.
result First estimation error bound for trace norm regularized estimator with scarce data.
New algorithms solve large-scale rank minimization problems efficiently.
problem Large-scale rank minimization problems.
method Define and apply bi-trace and tri-trace norms to rank minimization problems; design efficient linearized alternating minimization algorithms.
result Proved algorithms converge to critical points; provide RSC and MC error bounds.
Using the -norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, ca…
New pivoting strategy improves trace norm contraction in low-rank approximation.
problem Finding good low-rank approximations of symmetric, positive-definite matrices.
method Choosing rows with likelihood proportional to for randomly pivoted partial Cholesky algorithm.
result Same trace norm contraction result in Frobenius norm for improved pivoting strategy.
Unified formula for higher traces of linear maps on finite-dimensional normed spaces.
problem Unified trace-average formula for higher traces of linear maps.
method Unified trace-average formula for the k-th higher trace of a linear operator A on a finite-dimensional normed space.
result Unified trace-average formula holds for all A if and only if the operator-valued average equals the identity.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
Improved Frank-Wolfe algorithm solves convex trace-norm ball problems.
problem Optimizing convex functions over trace-norm balls.
method Rank-k variant of Frank-Wolfe algorithm using top-k singular-vector computation.
result Linear convergence rate for smooth and strongly convex objectives with rank-limited solutions.
A distributed algorithm for learning low-rank matrices from large datasets.
problem Learning high-dimensional low-rank matrices from distributed data with trace norm constraint.
method DFW-Trace, a distributed Frank-Wolfe algorithm using power method approximations.
result DFW-Trace achieves sublinear convergence to optimal solutions with few power iterations.
We provide rigorous guarantees on learning with the weighted trace-norm under arbitrary sampling distributions. We show that the standard weighted trace-norm might fail when the sampling distribution is not a product distribution (i.e. when row and column indexes are not selected independently), present a corrected var…
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…
Suppose a given observation matrix can be decomposed as the sum of a low-rank matrix and a sparse matrix (outliers), and the goal is to recover these individual components from the observed sum. Such additive decompositions have applications in a variety of numerical problems including system identification, latent var…
Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are independent of the dimension of the input space, which may be infinite as in the case o…
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 …
Spectral regularization simplifies sequence models by focusing on grammatical simplicity.
problem Sequence modeling challenges in learning tasks.
method Introduces spectral regularization based on Hankel matrices and trace norm, addressing bi-infinite matrices with an unbiased estimator.
result Demonstrates spectral regularization's potential benefits on Tomita grammars.
Algorithm leverages low-rank relations between surrogate tasks for structured prediction.
problem Structured prediction with large or infinite-dimensional surrogate spaces.
method Trace norm regularization to leverage relationships between surrogate outputs without explicit coding/decoding functions.
result Our algorithm can improve generalization performance over previous methods.
Study spectral properties of sparse random graphs to recover latent vectors.
problem Recovering latent vectors in sparse random geometric graphs.
method Analyzes spectral concentration and uses orthogonal polynomial expansions, decoupling, and matrix concentration.
result Sharpens spectral norm bounds and proves exact recovery for Gaussian mixture models.
Differentiable relaxation for inferring partial orders from noisy linear data.
problem Inference of partial orders from linear data with noisy observations.
method Introducing a differentiable relaxation to model noisy linear extensions, replacing discontinuous precedence and feasibility with smooth surrogates.
result Smooth posterior that preserves partial-order semantics, supports gradient-based inference, and converges to hard likelihood.
New techniques for faster and more compact speech recognition models.
problem Efficiency and compactness in speech recognition neural networks.
method Trace norm regularization for low rank factoring and ARM optimized kernels for faster inference.
result 3x to 7x speed up in inference on ARM processors compared to gemmlowp.
A new method for math reasoning that allows for iterative correction.
problem Standard reasoning models commit to each token and cannot recover from early errors.
method Generative framework with latent thought vectors for iterative self-correction.
result 30 rethinking iterations surpass baselines with 15 times more parameters.
A new method clusters multi-view data by sharing a common trace-norm of coefficient matrices.
problem Insufficient exploitation of multi-view data due to uniform coefficient matrices.
method Imposes bilinear factorization with orthonormality and low-rank constraints on coefficient matrices.
result The proposed CBF-MSC method effectively clusters multi-view data more comprehensively.
Paper extends principal component pursuit to hypercomplex numbers for improved audio data analysis.
problem Improving robust principal component analysis for audio data.
method Extends principal component pursuit to polar -complex and -bicomplex numbers, deriving proximity operators for - and trace-norm regularizers.
result Our approach outperforms tensor robust principal component analysis on audio data.
We study the problem of learning a tensor from a set of linear measurements. A prominent methodology for this problem is based on a generalization of trace norm regularization, which has been used extensively for learning low rank matrices, to the tensor setting. In this paper, we highlight some limitations of this app…
Thurston's norm revisitedmath.GT
We study the Thurston norm on the second homology of a 3-manifold M, which is the surface bundle over the circle with a pseudo-Anosov monodromy. A novelty of our approach consists in the application of the C*-algebras to a problem in topology. Namely, one associates to M a C*-algebra, whose K-theory gives rise to an al…
GL-LowPopArt improves minimax-optimal estimation for trace regression.
problem Minimizing estimation error in generalized low-rank trace regression.
method Two-stage approach: nuclear norm regularization followed by matrix Catoni estimation.
result Achieves instance-wise optimal error bounds up to condition number.
Paper tackles clipped matrix recovery from scientific areas with theoretical and practical methods.
problem Recovering low-rank matrices from clipped observations in scientific areas.
method Trace-norm minimization algorithm and squared hinge loss with a novel regularization term.
result Theoretical guarantee and practical algorithms for exact recovery of clipped matrix completion.
New adversarial examples with structured distortion sets improve robustness and perceptibility.
problem Improving adversarial robustness and perceptibility of images.
method Exploring and constraining adversarial search with trace-norms and other norms.
result Structured adversarial perturbations allow larger distortions and control over generation.
Study optimizes KSD estimation from samples, revealing Hilbert-Schmidt vs trace scales.
problem Optimizing estimation of Kernel Stein Discrepancy from samples.
method Identifying and comparing minimax scales for U-statistic and V-statistic.
result Hilbert-Schmidt norm of Stein covariance operator gives optimal scale.
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…
Paper proposes a framework for privacy-preserving machine learning.
problem Extracted traces from pervasive systems violate user privacy.
method Interpretable machine learning framework for privacy preservation.
result Users can understand how their privacy is compromised by machine learning.
We study the problem of estimating multiple predictive functions from a dictionary of basis functions in the nonparametric regression setting. Our estimation scheme assumes that each predictive function can be estimated in the form of a linear combination of the basis functions. By assuming that the coefficient matrix …
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 …
An important and natural question in the analysis of Ricci flow singularity formation in dimensions four and above is as follows: What are the weakest conditions that provide control of the norm of the Riemann curvature tensor? In this short note, we show that on a compact manifold, the trace-free Ricci tensor is contr…
We analyze the structure of covariance matrices under graph constraints.
problem Analyzing the structure of covariance matrices under graph constraints.
method We explore the algebraic structure of the solution space of convex optimization problem Constrained Minimum Trace Factor Analysis (CMTFA) under a latent star topology.
result CMTFA can have either a rank 1 or a rank n-1 solution, with conditions for both.
A new metric assesses latent variable models using data and model moments.
problem Difficulty in assessing the quality of unsupervised learning models.
method A moment-matching metric using matrix norms to compare data and model moments.
result The proposed metric is faster and has less variance than alternative methods.
CausalSim corrects bias in trace-driven simulations for more accurate results.
problem Bias in trace-driven simulations due to system conditions during trace collection.
method CausalSim learns a causal model of system dynamics and latent factors from an RCT to remove bias from trace data.
result CausalSim reduces simulation errors by 53% and 61% compared to baselines, providing more accurate insights.
Estimates latent norms and Gram matrices for graphs on Euclidean balls.
problem Estimating latent points and their relationships in graphs on Euclidean balls.
method Estimates latent norms and Gram matrices using observed graph data.
result Graphs on Euclidean balls can have power-law degree distributions.
In this paper, we propose an unifying view of several recently proposed structured sparsity-inducing norms. We consider the situation of a model simultaneously (a) penalized by a set- function de ned on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) regularized in Lp-n…
We study a norm for structured sparsity which leads to sparse linear predictors whose supports are unions of prede ned overlapping groups of variables. We call the obtained formulation latent group Lasso, since it is based on applying the usual group Lasso penalty on a set of latent variables. A detailed analysis of th…
The paper proves new curvature estimates in quaternionic contact geometry.
problem Estimating the curvature of compact qc manifolds.
method Establishing quaternionic contact versions of the Almost Schur Lemma.
result Curvature is a constant in terms of specific norms and components.
Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of nonzero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential ap…
If is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is …
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,…