The Association for Women in Mathematics (AWM) 2023 Research Symposium featured a Session on Tensor Methods for Data Modeling organized by Anna Konstorum (Center for Computing Sciences, Institute for Defense Analyses). The goal of the session was to bring together speakers working in on a variety of tensor analysis methods with the aim to improve their applicability to real-world problems. Slides shared here have been made available by the speakers.
Abstract
Research into applications of tensor decomposition methods to
real-world data has been accelerating, driven by the increasing
complexity of data sets arising from a broad range of fields, including
the biomedical, network, and physical and social sciences. Tensor
methods can aid in low-dimensional sample and feature embedding and
clustering, as well as data compression. Current challenges in
deployment of tensor decompositions for data analytics include questions
of existence and uniqueness of optimal solutions, rank and model
selection, computational complexity, and data imputation. The goal of
this session is to bring together speakers that develop and implement
tensor methods on real-world datasets, including aspects relating to the
theoretical and/or computational challenges, to foster collaboration and
discussion.
Talks
Hierarchical nonnegative tensor factorizations and applications
Speaker: Jamie Haddock (Harvey Mudd College)
Co-authors: Deanna Needell, Joshua Vendrow
Abstract. Nonnegative matrix factorization (NMF) has
found many applications including topic modeling and document analysis.
Hierarchical NMF (HNMF) variants are able to learn topics at various
levels of granularity and illustrate their hierarchical relationship.
Recently, nonnegative tensor factorization (NTF) methods have been
applied in a similar fashion in order to handle data sets with complex,
multi-modal structure. Hierarchical NTF (HNTF) methods have been
proposed, however these methods do not naturally generalize their
matrix-based counterparts. Here, we propose a new HNTF model which
directly generalizes a HNMF model special case, and provide a supervised
extension. Our experimental results show that this model more naturally
illuminates the topic hierarchy than previous HNMF and HNTF methods.
Recovering Noisy Tensors from Double Sketches
Speaker: Anna Ma (UC Irvine)
Co-authors: Yizhe
Zhu, Dominik Stoeger
Abstract. Machine learning tasks often utilize vast
amounts of data where data can be represented in various ways. Instead
of representing data as one or two-dimensional arrays, one can instead
use multi-dimensional arrays to describe more complex relationships
between elements in a data set. Moving beyond the matrix or vector
representation of data requires using tensors and tensor operations. In
this talk, we focus on using the tensor-tensor t-product for recovering
tensors of total and assumably low tensor rank. In particular, we will
explore using random Gaussian sketches for recovering tensors and
present provable guarantees for recovery.
Robust Tensor CUR: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptions
Speaker: Longxiu Huang (Michigan State University)
Co-authors: HanQin Cai, Zehan Chao, Longxiu Huang, and Deanna
Needell
Abstract. We study the problem of tensor robust
principal component analysis (TRPCA), which aims to separate an
underlying low-multilinear-rank tensor and a sparse outlier tensor from
their sum. This work proposes fast algorithms, called Robust Tensor
CUR(RTCUR), for large-scale non-convex TRPCA problems under the Tucker
rank setting. RTCUR is developed within a framework of alternating
projections that projects between the set of low-rank tensors and the
set of sparse tensors. We utilize the recently developed tensor CUR
decomposition to substantially reduce the computational complexity in
each projection. In addition, we develop four variants of RTCUR for
different application settings. We demonstrate the effectiveness and
computational advantages of RTCUR against state-of-the-art methods on
both synthetic and real-world datasets.
Tensor BM-Decomposition for Compression and Analysis of Spatio-Temporal Third-order Data
Speaker: Fan Tian (Tufts University)
Co-authors: Misha E. Kilmer, Eric Miller, Abani Patra
Abstract. We introduce a third-order tensor decomposition framework based on a ternary multiplication named Bhattacharya-Mesner (BM) product and its corresponding notion of rank. We describe an iterative algorithm for computing a low BM-rank approximation to a given third-order data array. Moreover, we will demonstrate our decomposition framework for video processing applications. Our method appears to improve on other matrix and tensor-based methods with smaller approximation error for the same level of compression.