Partial Trace Regression and Low-Rank Kraus Decomposition
Aix-Marseille University（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；LIS（计算机科学与系统实验室）；Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所）；Aalto University（阿尔托大学）；Université Rouen Normandie（鲁昂诺曼底大学）；LITIS（信息处理与系统实验室）；Criteo AI Lab（Criteo人工智能实验室）
The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs; this model subsumes the trace regression model and thus the linear regression model. Borrowing tools from quantum information theory, where partial trace operators have been extensively studied, we propose a framework for learning partial trace regression models from data by taking advantage of the so-called low-rank Kraus representation of completely positive maps. We show the relevance of our framework with synthetic and real-world experiments conducted for both i) matrix-to-matrix regression and ii) positive semidefinite matrix completion, two tasks which can be formulated as partial trace regression problems.
Over the past few years, trace regression models have received considerable attention in the context of matrix completion, quantum state tomography, and compressed sensing. Estimation of the underlying matrix from regularization-based approaches promoting low-rankedness, notably nuclear norm regularization, have enjoyed great popularity. In the present paper, we argue that such regularization may no longer be necessary if the underlying matrix is symmetric positive semidefinite (\textsf{spd}) and the design satisfies certain conditions. In this situation, simple least squares estimation subjec…
University of Wisconsin–Madison（威斯康星大学麦迪逊分校）；University of Arizona（亚利桑那大学）
Learning of matrix-valued data has recently surged in a range of scientific and business applications. Trace regression is a widely used method to model effects of matrix predictors and has shown great success in matrix learning. However, nearly all existing trace regression solutions rely on two assumptions: (i) a known functional form of the conditional mean, and (ii) a global low-rank structure in the entire range of the regression function, both of which may be violated in practice. In this article, we relax these assumptions by developing a general framework for nonparametric trace regres…
Matrix completion and quantum tomography are two unrelated research areas with great current interest in many modern scientific studies. This paper investigates the statistical relationship between trace regression in matrix completion and quantum state tomography in quantum physics and quantum information science. As quantum state tomography and trace regression share the common goal of recovering an unknown matrix, it is nature to put them in the Le Cam paradigm for statistical comparison. Regarding the two types of matrix inference problems as two statistical experiments, we establish their…
We propose trace pursuit for model-free variable selection under the sufficient dimension reduction paradigm. Two distinct algorithms are proposed: stepwise trace pursuit and forward trace pursuit. Stepwise trace pursuit achieves selection consistency with fixed p, and is readily applicable in the challenging setting with p>n. Forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality. The screening consistency property of forward trace pursuit based on sliced inverse regression is established. Finite sample performances of …
In the present paper we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs we obtain bounds for the minimax rates of certain composite hy…
Rescaled spike and slab models are a new Bayesian variable selection method for linear regression models. In high dimensional orthogonal settings such models have been shown to possess optimal model selection properties. We review background theory and discuss applications of rescaled spike and slab models to prediction problems involving orthogonal polynomials. We first consider global smoothing and discuss potential weaknesses. Some of these deficiencies are remedied by using local regression. The local regression approach relies on an intimate connection between local weighted regression an…
We study the problem of selecting features associated with extreme values in high dimensional linear regression. Normally, in linear modeling problems, the presence of abnormal extreme values or outliers is considered an anomaly which should either be removed from the data or remedied using robust regression methods. In many situations, however, the extreme values in regression modeling are not outliers but rather the signals of interest; consider traces from spiking neurons, volatility in finance, or extreme events in climate science, for example. In this paper, we propose a new method for sp…
A novel approach to malware classification is introduced based on analysis of instruction traces that are collected dynamically from the program in question. The method has been implemented online in a sandbox environment (i.e., a security mechanism for separating running programs) at Los Alamos National Laboratory, and is intended for eventual host-based use, provided the issue of sampling the instructions executed by a given process without disruption to the user can be satisfactorily addressed. The procedure represents an instruction trace with a Markov chain structure in which the transiti…
This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a new nuclear norm penalized estimator of $A_0$ and establish a general sharp oracle inequality for this estimator for arbitrary values of $n,m_1,m_2$ under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in t…
In this paper we consider the trace regression model. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A_0$ corrupted by noise. We propose a new rank penalized estimator of $A_0$. For this estimator we establish general oracle inequality for the prediction error both in probability and in expectation. We also prove upper bounds for the rank of our estimator. Then, we apply our general results to the problems of matrix completion and matrix regression. In these cases our estimator has a particularly simple form: it is obtained by hard thresho…
Tokyo Institute of Technology（东京工业大学）；The Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；University of Tokyo（东京大学）
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 of multipliers, which is computationally efficient when the number of training samples is moderate. We then theoretically derive an excess risk bound for each tensor norm and clarify their behavior. Finally, we perform extensive experiments using simulated and real data and demon…
Many problems in signal processing require finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. When dealing with real-world data, the presence of outliers and impulsive noise must also be accounted for. In past decades, the vast majority of robust linear regression estimators has focused on robustness against rowwise contamination. Even so called `high breakdown' estimators rely on the assumption that a majority of rows of the regression matrix is not affected by outliers. Only very recently, the first cellwise robust regression estimation methods hav…
This paper introduces a simple principle for robust high-dimensional statistical inference via an appropriate shrinkage on the data. This widens the scope of high-dimensional techniques, reducing the moment conditions from sub-exponential or sub-Gaussian distributions to merely bounded second or fourth moment. As an illustration of this principle, we focus on robust estimation of the low-rank matrix $Θ^*$ from the trace regression model $Y=Tr (Θ^{*T}X) +ε$. It encompasses four popular problems: sparse linear models, compressed sensing, matrix completion and multi-task regression. We propose to…
Weill Cornell Medicine（威尔康奈尔医学院）；University of Connecticut（康涅狄格大学）；Michigan State Universtiy（密歇根州立大学）
We propose a sparse and low-rank tensor regression model to relate a univariate outcome to a feature tensor, in which each unit-rank tensor from the CP decomposition of the coefficient tensor is assumed to be sparse. This structure is both parsimonious and highly interpretable, as it implies that the outcome is related to the features through a few distinct pathways, each of which may only involve subsets of feature dimensions. We take a divide-and-conquer strategy to simplify the task into a set of sparse unit-rank tensor regression problems. To make the computation efficient and scalable, fo…
We propose a novel hierarchical model for multitask bipartite ranking. The proposed approach combines a matrix-variate Gaussian process with a generative model for task-wise bipartite ranking. In addition, we employ a novel trace constrained variational inference approach to impose low rank structure on the posterior matrix-variate Gaussian process. The resulting posterior covariance function is derived in closed form, and the posterior mean function is the solution to a matrix-variate regression with a novel spectral elastic net regularizer. Further, we show that variational inference for the…
We propose the convex factorization machine (CFM), which is a convex variant of the widely used Factorization Machines (FMs). Specifically, we employ a linear+quadratic model and regularize the linear term with the $\ell_2$-regularizer and the quadratic term with the trace norm regularizer. Then, we formulate the CFM optimization as a semidefinite programming problem and propose an efficient optimization procedure with Hazan's algorithm. A key advantage of CFM over existing FMs is that it can find a globally optimal solution, while FMs may get a poor locally optimal solution since the objectiv…
Department of Electrical Engineering and Computer Science, University of Liège（列日大学电气工程与计算机科学系）；INRIA - Sierra Project-Team（法国国家信息与自动化研究所 - Sierra项目组）
The paper addresses the problem of low-rank trace norm minimization. We propose an algorithm that alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is characterized by an efficient factorization that makes the trace norm differentiable in the search space and the computation of duality gap numerically tractable. The search space is nonlinear but is equipped with a particular Riemannian structure that leads to efficient computations. We present a second-order trust-region algorithm with a guaranteed quadratic rate of convergence. Overall, the proposed …
ENS Paris-Saclay（巴黎萨克雷高等师范学院）；Inria（法国国家信息与自动化研究所）；Deepmind（深度思维）；ENSAE（法国国家统计与经济管理学院）；Criteo AI Lab（Criteo人工智能实验室）
We consider in this paper the problem of optimal experiment design where a decision maker can choose which points to sample to obtain an estimate $\hatβ$ of the hidden parameter $β^{\star}$ of an underlying linear model. The key challenge of this work lies in the heteroscedasticity assumption that we make, meaning that each covariate has a different and unknown variance. The goal of the decision maker is then to figure out on the fly the optimal way to allocate the total budget of $T$ samples between covariates, as sampling several times a specific one will reduce the variance of the estimated…
