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Fan Tian

Publications and source records attributed to Fan Tian.

5 recordsLinked to original sources
arXiv2025

A tensor-based dynamic mode decomposition based on the $\star_{\boldsymbol{M}}$-product

Dynamic mode decomposition (DMD) is a data-driven method for estimating the dynamics of a discrete dynamical system. This paper proposes a tensor-based approach to DMD for applications in which the states can be viewed as tensors. Specifically, we use the $\star_{\boldsymbol{M}}$-product framework for tensor decompositions which we demonstrate offers excellent compression compared to matrix-based methods and can be implemented in a computationally efficient manner. We show how the proposed approach is connected to the traditional DMD and physics-informed DMD frameworks. We give a computational framework for computing the tensor-based DMD and detail the computational costs. We also give a randomized algorithm that enables efficient $\star_{\boldsymbol{M}}$-DMD computations in the streaming setting. The numerical results show that the proposed method achieves equal or better accuracy for the same storage compared to the standard DMD on these examples and is more efficient to compute.

math.NA↗
arXiv2024

Tensor Completion with BMD Factor Nuclear Norm Minimization

This paper is concerned with the problem of recovering third-order tensor data from limited samples. A recently proposed tensor decomposition (BMD) method has been shown to efficiently compress third-order spatiotemporal data. Using the BMD, we formulate a slicewise nuclear norm penalized algorithm to recover a third-order tensor from limited observed samples. We develop an efficient alternating direction method of multipliers (ADMM) scheme to solve the resulting minimization problem. Experimental results on real data show our method to give reconstruction comparable to those of HaLRTC (Liu et al., IEEE Trans Ptrn Anal Mchn Int, 2012), a well-known tensor completion method, in about the same number of iterations. However, our method has the advantage of smaller subproblems and higher parallelizability per iteration.

math.NA↗
arXiv2023

Tensor BM-Decomposition for Compression and Analysis of Video Data

Given tensors $\boldsymbol{\mathscr{A}}, \boldsymbol{\mathscr{B}}, \boldsymbol{\mathscr{C}}$ of size $m \times 1 \times n$, $m \times p \times 1$, and $1\times p \times n$, respectively, their Bhattacharya-Mesner (BM) product will result in a third-order tensor of dimension $m \times p \times n$ and BM-rank of 1 (Mesner and Bhattacharya, 1990). Thus, if an arbitrary $m \times p \times n$ third-order tensor can be written as a sum of a small number, relative to $m,p,n$, of such BM-rank 1 terms, this BM-decomposition (BMD) offers an implicitly compressed representation of the tensor. In this paper, we first show that grayscale surveillance video can be accurately captured by a low BM-rank decomposition and give methods for efficiently computing this decomposition. To this end, we first give results that connect rank-revealing matrix factorizations to the BMD. Next, we present a generative model that illustrates that spatio-temporal video data can be expected to have low BM-rank. We combine these observations to derive a regularized alternating least squares (ALS) algorithm to compute an approximate BMD of the video tensor. The algorithm itself is highly parallelizable since the bulk of the computations break down into relatively small regularized least squares problems that can be solved independently. Extensive numerical results compared against the state-of-the-art matrix-based DMD for surveillance video separation show our algorithms can consistently produce results with superior compression properties while simultaneously providing better separation of stationary and non-stationary features in the data. We then introduce a new type of BM-product suitable for color video and provide an algorithm that shows an impressive ability to extract important temporal information from color video while simultaneously compressing the data.

math.NA↗
arXiv2019

Robust Registration of Astronomy Catalogs with Applications to the Hubble Space Telescope

Astrometric calibration of images with a small field of view is often inferior to the internal accuracy of the source detections due to the small number of accessible guide stars. One important experiment with such challenges is the Hubble Space Telescope (HST). A possible solution is to cross-calibrate overlapping fields instead of just relying on standard stars. Following the approach of \citet{2012ApJ...761..188B}, we use infinitesimal 3D rotations for fine-tuning the calibration but devise a better objective that is robust to a large number of false candidates in the initial set of associations. Using Bayesian statistics, we accommodate bad data by explicitly modeling the quality, which yields a formalism essentially identical to an $M$-estimation in robust statistics. Our results on simulated and real catalogs show great potentials for improving the HST calibration, and those with similar challenges.

astro-ph.IM↗