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Register a new userSpectral Algorithm for Low-rank Multitask Regression
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Multitask learning, i.e. taking advantage of the relatedness of individual tasks in order to improve performance on all of them, is a core challenge in the field of machine learning. We focus on matrix regression tasks where the rank of the weight matrix is constrained to reduce sample complexity. We introduce the common mechanism regression (CMR) model which assumes a shared left low-rank component across all tasks, but allows an individual per-task right low-rank component. This dramatically reduces the number of samples needed for accurate estimation. The problem of jointly recovering the common and the local components has a non-convex bi-linear structure. We overcome this hurdle and provide a provably beneficial non-iterative spectral algorithm. Appealingly, the solution has favorable behavior as a function of the number of related tasks and the small number of samples available for each one. We demonstrate the efficacy of our approach for the challenging task of remote river discharge estimation across multiple river sites, where data for each task is naturally scarce. In this scenario sharing a low-rank component between the tasks translates to a shared spectral reflection of the water, which is a true underlying physical model. We also show the benefit of the approach on the markedly different setting of image classification where the common component can be interpreted as the shared convolution filters.
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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, for the unit-rank tensor regression, we propose a stagewise estimation procedure to efficiently trace out its entire solution path. We show that as the step size goes to zero, the stagewise solution paths converge exactly to those of the corresponding regularized regression. The superior performance of our approach is demonstrated on various real-world and synthetic examples.
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 also called tensor grid (TG), allows more interactions between different dimensions, and may lead to more compact representation. In this paper, we propose to perform image completion based on low-rank tensor grid. A two-stage density matrix renormalization group algorithm is used for initialization of TG decomposition, which consists of multiple TT decompositions. The latent TG factors can be alternatively obtained by solving alternating least squares problems. To further improve the computational efficiency, a multi-linear matrix factorization for low rank TG completion is developed by using parallel matrix factorization. Experimental results on synthetic data and real-world images show the proposed methods outperform the existing ones in terms of recovery accuracy.
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.
This paper considers the problem of matrix-variate logistic regression. The fundamental error threshold on estimating coefficient matrices in the logistic regression problem is found by deriving a lower bound on the minimax risk. The focus of this paper is on derivation of a minimax risk lower bound for low-rank coefficient matrices. The bound depends explicitly on the dimensions and distribution of the covariates, the rank and energy of the coefficient matrix, and the number of samples. The resulting bound is proportional to the intrinsic degrees of freedom in the problem, which suggests the sample complexity of the low-rank matrix logistic regression problem can be lower than that for vectorized logistic regression. color{red}color{black} The proof techniques utilized in this work also set the stage for development of minimax lower bounds for tensor-variate logistic regression problems.
We present our entry into the 2021 3C Shared Task Citation Context Classification based on Purpose competition. The goal of the competition is to classify a citation in a scientific article based on its purpose. This task is important because it could potentially lead to more comprehensive ways of summarizing the purpose and uses of scientific articles, but it is also difficult, mainly due to the limited amount of available training data in which the purposes of each citation have been hand-labeled, along with the subjectivity of these labels. Our entry in the competition is a multi-task model that combines multiple modules designed to handle the problem from different perspectives, including hand-generated linguistic features, TF-IDF features, and an LSTM-with-attention model. We also provide an ablation study and feature analysis whose insights could lead to future work.
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