Low-rank approximation of tensors via sparse optimization

Complex data analytics via sparse, low-rank matrix approximation

Sparse Principal Component Analysis via Regularized Low Rank Matrix Approximation

Principal component analysis (PCA) is a widely used tool for data analysis and dimension reduction in applications throughout science and engineering. However, the principal components (PCs) can sometimes be difficult to interpret, because they are linear combinations of all the original variables. To facilitate interpretation, sparse PCA produces modified PCs with sparse loadings, i.e. loading...

Embedding Lexical Features via Low-Rank Tensors

Modern NLP models rely heavily on engineered features, which often combine word and contextual information into complex lexical features. Such combination results in large numbers of features, which can lead to overfitting. We present a new model that represents complex lexical features—comprised of parts for words, contextual information and labels—in a tensor that captures conjunction informa...

Low-rank Matrices in the Approximation of Tensors

Most successful numerical algorithms for multi-dimensional problems usually involve multi-index arrays, also called tensors, and capitalize on those tensor decompositions that reduce, one way or another, to low-rank matrices associated with the given tensors. It can be argued that the most of recent progress is due to the TT and HT decompostions [1]. The differences between the two decompositio...

Low-Rank Approximation and Completion of Positive Tensors

Unlike the matrix case, computing low-rank approximations of tensors is NP-hard and numerically ill-posed in general. Even the best rank-1 approximation of a tensor is NP-hard. In this paper, we use convex optimization to develop polynomial-time algorithms for low-rank approximation and completion of positive tensors. Our approach is to use algebraic topology to define a new (numerically well-p...