arXiv subjects

Deanna Needell

Publications and source records attributed to Deanna Needell.

At least 91 records · Page 5Linked to original sources

arXiv2022

Analysis of Spatial and Spatiotemporal Anomalies Using Persistent Homology: Case Studies with COVID-19 Data

We develop a method for analyzing spatial and spatiotemporal anomalies in geospatial data using topological data analysis (TDA). To do this, we use persistent homology (PH), which allows one to algorithmically detect geometric voids in a data set and quantify the persistence of such voids. We construct an efficient filtered simplicial complex (FSC) such that the voids in our FSC are in one-to-one correspondence with the anomalies. Our approach goes beyond simply identifying anomalies; it also encodes information about the relationships between anomalies. We use vineyards, which one can interpret as time-varying persistence diagrams (which are an approach for visualizing PH), to track how the locations of the anomalies change with time. We conduct two case studies using spatially heterogeneous COVID-19 data. First, we examine vaccination rates in New York City by zip code at a single point in time. Second, we study a year-long data set of COVID-19 case rates in neighborhoods of the city of Los Angeles.

physics.soc-ph↗

arXiv2022

A Generalized Hierarchical Nonnegative Tensor Decomposition

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. We also provide a multiplicative updates training method for this model. Our experimental results show that this model more naturally illuminates the topic hierarchy than previous HNMF and HNTF methods.

cs.LG↗

arXiv2022

Selectable Set Randomized Kaczmarz

The Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz (SSRK) is an variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive "selectable set" and thus yields an improved convergence guarantee. In this paper, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.

math.NA↗

arXiv2022

Guided Semi-Supervised Non-negative Matrix Factorization on Legal Documents

Classification and topic modeling are popular techniques in machine learning that extract information from large-scale datasets. By incorporating a priori information such as labels or important features, methods have been developed to perform classification and topic modeling tasks; however, most methods that can perform both do not allow for guidance of the topics or features. In this paper, we propose a method, namely Guided Semi-Supervised Non-negative Matrix Factorization (GSSNMF), that performs both classification and topic modeling by incorporating supervision from both pre-assigned document class labels and user-designed seed words. We test the performance of this method through its application to legal documents provided by the California Innocence Project, a nonprofit that works to free innocent convicted persons and reform the justice system. The results show that our proposed method improves both classification accuracy and topic coherence in comparison to past methods like Semi-Supervised Non-negative Matrix Factorization (SSNMF) and Guided Non-negative Matrix Factorization (Guided NMF).

cs.LG↗

arXiv2021

Analysis of Legal Documents via Non-negative Matrix Factorization Methods

The California Innocence Project (CIP), a clinical law school program aiming to free wrongfully convicted prisoners, evaluates thousands of mails containing new requests for assistance and corresponding case files. Processing and interpreting this large amount of information presents a significant challenge for CIP officials, which can be successfully aided by topic modeling techniques.In this paper, we apply Non-negative Matrix Factorization (NMF) method and implement various offshoots of it to the important and previously unstudied data set compiled by CIP. We identify underlying topics of existing case files and classify request files by crime type and case status (decision type). The results uncover the semantic structure of current case files and can provide CIP officials with a general understanding of newly received case files before further examinations. We also provide an exposition of popular variants of NMF with their experimental results and discuss the benefits and drawbacks of each variant through the real-world application.

cs.LG↗

arXiv2021

On audio enhancement via online non-negative matrix factorization

We propose a method for noise reduction, the task of producing a clean audio signal from a recording corrupted by additive noise. Many common approaches to this problem are based upon applying non-negative matrix factorization to spectrogram measurements. These methods use a noiseless recording, which is believed to be similar in structure to the signal of interest, and a pure-noise recording to learn dictionaries for the true signal and the noise. One may then construct an approximation of the true signal by projecting the corrupted recording on to the clean dictionary. In this work, we build upon these methods by proposing the use of \emph{online} non-negative matrix factorization for this problem. This method is more memory efficient than traditional non-negative matrix factorization and also has potential applications to real-time denoising.

eess.AS↗

arXiv2021

Robust recovery of bandlimited graph signals via randomized dynamical sampling

Heat diffusion processes have found wide applications in modelling dynamical systems over graphs. In this paper, we consider the recovery of a $k$-bandlimited graph signal that is an initial signal of a heat diffusion process from its space-time samples. We propose three random space-time sampling regimes, termed dynamical sampling techniques, that consist in selecting a small subset of space-time nodes at random according to some probability distribution. We show that the number of space-time samples required to ensure stable recovery for each regime depends on a parameter called the spectral graph weighted coherence, that depends on the interplay between the dynamics over the graphs and sampling probability distributions. In optimal scenarios, no more than $\mathcal{O}(k \log(k))$ space-time samples are sufficient to ensure accurate and stable recovery of all $k$-bandlimited signals. In any case, dynamical sampling typically requires much fewer spatial samples than the static case by leveraging the temporal information. Then, we propose a computationally efficient method to reconstruct $k$-bandlimited signals from their space-time samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we test dynamical sampling techniques on a wide variety of graphs. The numerical results support our theoretical findings and demonstrate the efficiency.

cs.IT↗

arXiv2021

Modewise Operators, the Tensor Restricted Isometry Property, and Low-Rank Tensor Recovery

Recovery of sparse vectors and low-rank matrices from a small number of linear measurements is well-known to be possible under various model assumptions on the measurements. The key requirement on the measurement matrices is typically the restricted isometry property, that is, approximate orthonormality when acting on the subspace to be recovered. Among the most widely used random matrix measurement models are (a) independent sub-gaussian models and (b) randomized Fourier-based models, allowing for the efficient computation of the measurements. For the now ubiquitous tensor data, direct application of the known recovery algorithms to the vectorized or matricized tensor is awkward and memory-heavy because of the huge measurement matrices to be constructed and stored. In this paper, we propose modewise measurement schemes based on sub-gaussian and randomized Fourier measurements. These modewise operators act on the pairs or other small subsets of the tensor modes separately. They require significantly less memory than the measurements working on the vectorized tensor, provably satisfy the tensor restricted isometry property and experimentally can recover the tensor data from fewer measurements and do not require impractical storage.

math.NA↗

arXiv2021

Fast Robust Tensor Principal Component Analysis via Fiber CUR Decomposition

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. In this work, we propose a fast non-convex algorithm, coined Robust Tensor CUR (RTCUR), for large-scale TRPCA problems. RTCUR considers a framework of alternating projections and utilizes the recently developed tensor Fiber CUR decomposition to dramatically lower the computational complexity. The performance advantage of RTCUR is empirically verified against the state-of-the-arts on the synthetic datasets and is further demonstrated on the real-world application such as color video background subtraction.

cs.LG↗

arXiv2021

Robust CUR Decomposition: Theory and Imaging Applications

This paper considers the use of Robust PCA in a CUR decomposition framework and applications thereof. Our main algorithms produce a robust version of column-row factorizations of matrices $\mathbf{D}=\mathbf{L}+\mathbf{S}$ where $\mathbf{L}$ is low-rank and $\mathbf{S}$ contains sparse outliers. These methods yield interpretable factorizations at low computational cost, and provide new CUR decompositions that are robust to sparse outliers, in contrast to previous methods. We consider two key imaging applications of Robust PCA: video foreground-background separation and face modeling. This paper examines the qualitative behavior of our Robust CUR decompositions on the benchmark videos and face datasets, and find that our method works as well as standard Robust PCA while being significantly faster. Additionally, we consider hybrid randomized and deterministic sampling methods which produce a compact CUR decomposition of a given matrix, and apply this to video sequences to produce canonical frames thereof.

cs.CV↗

arXiv2021

QuantileRK: Solving Large-Scale Linear Systems with Corrupted, Noisy Data

Measurement data in linear systems arising from real-world applications often suffers from both large, sparse corruptions, and widespread small-scale noise. This can render many popular solvers ineffective, as the least squares solution is far from the desired solution, and the underlying consistent system becomes harder to identify and solve. QuantileRK is a member of the Kaczmarz family of iterative projective methods that has been shown to converge exponentially for systems with arbitrarily large sparse corruptions. In this paper, we extend the analysis to the case where there are not only corruptions present, but also noise that may affect every data point, and prove that QuantileRK converges with the same rate up to an error threshold. We give both theoretical and experimental results demonstrating QuantileRK's strength.

math.NA↗

arXiv2021

Quantile-based Iterative Methods for Corrupted Systems of Linear Equations

Often in applications ranging from medical imaging and sensor networks to error correction and data science (and beyond), one needs to solve large-scale linear systems in which a fraction of the measurements have been corrupted. We consider solving such large-scale systems of linear equations $\mathbf{A}\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in the measurement vector $\mathbf{b}$. We develop several variants of iterative methods that converge to the solution of the uncorrupted system of equations, even in the presence of large corruptions. These methods make use of a quantile of the absolute values of the residual vector in determining the iterate update. We present both theoretical and empirical results that demonstrate the promise of these iterative approaches.

math.NA↗

arXiv2021

HOSVD-Based Algorithm for Weighted Tensor Completion

Matrix completion, the problem of completing missing entries in a data matrix with low dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog, that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted HOSVD algorithm for recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations.

math.NA↗

arXiv2021

Mode-wise Tensor Decompositions: Multi-dimensional Generalizations of CUR Decompositions

Low rank tensor approximation is a fundamental tool in modern machine learning and data science. In this paper, we study the characterization, perturbation analysis, and an efficient sampling strategy for two primary tensor CUR approximations, namely Chidori and Fiber CUR. We characterize exact tensor CUR decompositions for low multilinear rank tensors. We also present theoretical error bounds of the tensor CUR approximations when (adversarial or Gaussian) noise appears. Moreover, we show that low cost uniform sampling is sufficient for tensor CUR approximations if the tensor has an incoherent structure. Empirical performance evaluations, with both synthetic and real-world datasets, establish the speed advantage of the tensor CUR approximations over other state-of-the-art low multilinear rank tensor approximations.

math.NA↗

arXiv2021

Statistical Learning for Best Practices in Tattoo Removal

The causes behind complications in laser-assisted tattoo removal are currently not well understood, and in the literature relating to tattoo removal the emphasis on removal treatment is on removal technologies and tools, not best parameters involved in the treatment process. Additionally, the very challenge of determining best practices is difficult given the complexity of interactions between factors that may correlate to these complications. In this paper we apply a battery of classical statistical methods and techniques to identify features that may be closely correlated to causes of complication during the tattoo removal process, and report quantitative evidence for potential best practices. We develop elementary statistical descriptions of tattoo data collected by the largest gang rehabilitation and reentry organization in the world, Homeboy Industries; perform parametric and nonparametric tests of significance; and finally, produce a statistical model explaining treatment parameter interactions, as well as develop a ranking system for treatment parameters utilizing bootstrapping and gradient boosting.

stat.AP↗

arXiv2021

An Adaptation for Iterative Structured Matrix Completion

The task of predicting missing entries of a matrix, from a subset of known entries, is known as \textit{matrix completion}. In today's data-driven world, data completion is essential whether it is the main goal or a pre-processing step. Structured matrix completion includes any setting in which data is not missing uniformly at random. In recent work, a modification to the standard nuclear norm minimization (NNM) for matrix completion has been developed to take into account \emph{sparsity-based} structure in the missing entries. This notion of structure is motivated in many settings including recommender systems, where the probability that an entry is observed depends on the value of the entry. We propose adjusting an Iteratively Reweighted Least Squares (IRLS) algorithm for low-rank matrix completion to take into account sparsity-based structure in the missing entries. We also present an iterative gradient-projection-based implementation of the algorithm that can handle large-scale matrices. Finally, we present a robust array of numerical experiments on matrices of varying sizes, ranks, and level of structure. We show that our proposed method is comparable with the adjusted NNM on small-sized matrices, and often outperforms the IRLS algorithm in structured settings on matrices up to size $1000 \times 1000$.

math.NA↗

arXiv2021

On a Guided Nonnegative Matrix Factorization

Fully unsupervised topic models have found fantastic success in document clustering and classification. However, these models often suffer from the tendency to learn less-than-meaningful or even redundant topics when the data is biased towards a set of features. For this reason, we propose an approach based upon the nonnegative matrix factorization (NMF) model, deemed \textit{Guided NMF}, that incorporates user-designed seed word supervision. Our experimental results demonstrate the promise of this model and illustrate that it is competitive with other methods of this ilk with only very little supervision information.

cs.LG↗

arXiv2020

Applications of Online Nonnegative Matrix Factorization to Image and Time-Series Data

Online nonnegative matrix factorization (ONMF) is a matrix factorization technique in the online setting where data are acquired in a streaming fashion and the matrix factors are updated each time. This enables factor analysis to be performed concurrently with the arrival of new data samples. In this article, we demonstrate how one can use online nonnegative matrix factorization algorithms to learn joint dictionary atoms from an ensemble of correlated data sets. We propose a temporal dictionary learning scheme for time-series data sets, based on ONMF algorithms. We demonstrate our dictionary learning technique in the application contexts of historical temperature data, video frames, and color images.

cs.LG↗