Tensors, also known as multiway arrays, have become ubiquitous as representations for high dimensional operators or as convenient schemes for storing data. Tensor decompositions can be instrumental in revealing latent correlations residing in high dimensional spaces. Inconsistencies between tensor and matrix algebra complicate their straightforward applicability to problems in machine learning, recognition, and imaging. Researchers seeking to overcome those discrepancies have introduced several different candidate extensions, each introducing unique advantages and challenges. In this talk, we shall review some of the common tensor algebra definitions, discuss their limitations, and introduce a family of tensor algebras based on a new definition of tensor-tensor products. We then show how our framework permits the elegant extension of linear algebraic concepts and algorithms to tensors. In particular, we show the value of the matrix-mimetic properties of our tensor algebra in generalizing traditional matrix-based algorithms for compression, analysis, and learning. We provide several illustrations of our tensor framework in practice, including image recognition, video completion, and dynamic graph learning for prediction and classification.
Bio: Misha Elena Kilmer is the William Walker Professor of Mathematics at Tufts University. She has a secondary appointment in the Department of Computer Science at Tufts University and is a co-PI of Tufts TRIPODS Institute. Beginning July 2021, she began to serve concurrently as Deputy Director of the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University. Professor Kilmer is the 2023 recipient of the Tufts Distinguished Scholar Award. She was a Kirk Distinguished Visiting Fellow at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, in Spring 2023. In 2019, Prof. Kilmer was named a Fellow of the Society for Industrial and Applied Mathematics (SIAM) "for her fundamental contributions to numerical linear algebra and scientific computing, including ill-posed problems, tensor decompositions, and iterative methods.”