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This paper studies the Schatten-q error of low-rank matrix estimation by singular value decomposition under perturbation. We specifically establish a perturbation bound on the low-rank matrix estimation via a perturbation projection error bound. Then, we establish lower bounds to justify the tightness of the upper bound on the low-rank matrix estimation error. We further develop a user-friendly sinΘ bound for singular subspace perturbation based on the matrix perturbation projection error bound. Finally, we demonstrate the advantage of our results over the ones in the literature by simulation.

Summary In microbiome and genomic studies, the regression of compositional data has been a crucial tool for identifying microbial taxa or genes that are associated with clinical phenotypes. To account for the variation in sequencing depth, the classic log-contrast model is often used where read counts are normalized into compositions. However, zero read counts and the randomness in covariates remain critical issues. We introduce a surprisingly simple, interpretable and efficient method for the estimation of compositional data regression through the lens of a novel high-dimensional log-error-in-variable regression model. The proposed method provides corrections on sequencing data with possible overdispersion and simultaneously avoids any subjective imputation of zero read counts. We provide theoretical justifications with matching upper and lower bounds for the estimation error. The merit of the procedure is illustrated through real data analysis and simulation studies.

In mixed longitudinal studies, a group of subjects enter the study at different ages (cross-sectional) and are followed for successive years (longitudinal). In the context of such studies, we consider nonparametric covariance estimation with samples of noisy and partially observed functional trajectories. The proposed algorithm is based on a noniterative sequential-aggregation scheme with only basic matrix operations and closed-form solutions in each step. The good performance of the proposed method is supported by both theory and numerical experiments. We also apply the proposed procedure to a study on the working memory of midlife women, based on data from the Study of Women's Health Across the Nation (SWAN).

Rapid and accurate detection can help optimize patient treatment and improve infection control against nosocomial carbapenemase-producing organisms (CPO). In this study, a total of 217 routine clinical isolates (Enterobacterales and A. baumannii), including 178 CPOs and 39 non-CPOs, were tested to evaluate the performance of six phenotypic carbapenemase detection and classification assays, i.e., BD Phoenix CPO detect panel, Rapidec Carba-NP, O.K.N detection kit, and three carbapenem inactivation methods (CIMs; mCIM, eCIM, sCIM). The overall detection sensitivity and specificity were 98.78% (95.21–99.79%) and 79.49% (63.06–90.13%), respectively, for the BD phoenix CPO P/N test; 91.93% (86.30–95.45%) and 100% (88.83–100%), respectively, for the Rapidec Carba-NP; 98.06% (94.00–99.50%) and 97.44% (84.92–99.87%), respectively, for mCIM; and 96.89% (92.52–98.85%) and 94.87% (81.37–99.11%), respectively, for sCIM. The classification sensitivity and specificity for the BD phoenix CPO Ambler test, the O.K.N detection kit, and the mCIM and eCIM were 56.71% (48.75–64.34%) and 94.87% (81.37–99.11%), 99.28% (95.43–99.96%) and 100% (88.83–100%), and 92.90% (87.35–96.23%) and 97.44% (84.92–99.87%), respectively. All detection assays were reliable in detecting carbapenemase. However, the Rapidec Carba-NP and mCIM were insufficient in detecting OXA-48-like enzymes. The BD phoenix CPO detect panel had a strong ability to detect carbapenemase but failed to classify 48/59 (81.36%) KPC, 8/52 (15.38%) NDM, 8/22 (36.36%) OXA-23-like, and 6/11 (54.55%) dual enzymes. The O.K.N detection kit accurately detected and differentiated KPC, NDM, and OXA-48-like enzymes existing alone or in combination. The results of this study will support reliable laboratory work tools and promote therapeutic and infection control decisions.

Summary Metagenomics sequencing is routinely applied to quantify bacterial abundances in microbiome studies, where bacterial composition is estimated based on the sequencing read counts. Due to limited sequencing depth and DNA dropouts, many rare bacterial taxa might not be captured in the final sequencing reads, which results in many zero counts. Naive composition estimation using count normalization leads to many zero proportions, which tend to result in inaccurate estimates of bacterial abundance and diversity. This paper takes a multisample approach to estimation of bacterial abundances in order to borrow information across samples and across species. Empirical results from real datasets suggest that the composition matrix over multiple samples is approximately low rank, which motivates a regularized maximum likelihood estimation with a nuclear norm penalty. An efficient optimization algorithm using the generalized accelerated proximal gradient and Euclidean projection onto simplex space is developed. Theoretical upper bounds and the minimax lower bounds of the estimation errors, measured by the Kullback–Leibler divergence and the Frobenius norm, are established. Simulation studies demonstrate that the proposed estimator outperforms the naive estimators. The method is applied to an analysis of a human gut microbiome dataset.

In this article, we consider the sparse tensor singular value decomposition, which aims for dimension reduction on high-dimensional high-order data with certain sparsity structure. A method named Sparse Tensor Alternating Thresholding for Singular Value Decomposition (STAT-SVD) is proposed. The proposed procedure features a novel double projection \& thresholding scheme, which provides a sharp criterion for thresholding in each iteration. Compared with regular tensor SVD model, STAT-SVD permits more robust estimation under weaker assumptions. Both the upper and lower bounds for estimation accuracy are developed. The proposed procedure is shown to be minimax rate-optimal in a general class of situations. Simulation studies show that STAT-SVD performs well under a variety of configurations. We also illustrate the merits of the proposed procedure on a longitudinal tensor dataset on European country mortality rates., 73 pages

The transition kernel of a continuous-state-action Markov decision process (MDP) admits a natural tensor structure. This paper proposes a tensor-inspired unsupervised learning method to identify meaningful low-dimensional state and action representations from empirical trajectories. The method exploits the MDP's tensor structure by kernelization, importance sampling and low-Tucker-rank approximation. This method can be further used to cluster states and actions respectively and find the best discrete MDP abstraction. We provide sharp statistical error bounds for tensor concentration and the preservation of diffusion distance after embedding.

In this paper, we propose a general framework for sparse and low-rank tensor estimation from cubic sketchings. A two-stage non-convex implementation is developed based on sparse tensor decomposition and thresholded gradient descent, which ensures exact recovery in the noiseless case and stable recovery in the noisy case with high probability. The non-asymptotic analysis sheds light on an interplay between optimization error and statistical error. The proposed procedure is shown to be rate-optimal under certain conditions. As a technical by-product, novel high-order concentration inequalities are derived for studying high-moment sub-Gaussian tensors. An interesting tensor formulation illustrates the potential application to high-order interaction pursuit in high-dimensional linear regression., Accepted at IEEE Transactions on Information Theory

Model reduction of Markov processes is a basic problem in modeling state-transition systems. Motivated by the state aggregation approach rooted in control theory, we study the statistical state compression of a discrete-state Markov chain from empirical trajectories. Through the lens of spectral decomposition, we study the rank and features of Markov processes, as well as properties like representability, aggregability, and lumpability. We develop spectral methods for estimating the transition matrix of a low-rank Markov model, estimating the leading subspace spanned by Markov features, and recovering latent structures like state aggregation and lumpable partition of the state space. We prove statistical upper bounds for the estimation errors and nearly matching minimax lower bounds. Numerical studies are performed on synthetic data and a dataset of New York City taxi trips., Comment: to appear in IEEE Transactions on Information Theory

OBJECTIVE Postoperative hydrocephalus occurs in one-third of children after posterior fossa tumor resection. Although models to predict the need for CSF diversion after resection exist for preoperative variables, it is unknown which postoperative variables predict the need for CSF diversion. In this study, the authors sought to determine the clinical and radiographic predictors for CSF diversion in children following posterior fossa tumor resection. METHODS This was a retrospective cohort study involving patients ≤ 18 years of age who underwent resection of a primary posterior fossa tumor between 2000 and 2018. The primary outcome was the need for CSF diversion 6 months after surgery. Candidate predictors for CSF diversion including age, race, sex, frontal occipital horn ratio (FOHR), tumor type, tumor volume and location, transependymal edema, papilledema, presence of postoperative intraventricular blood, and residual tumor were evaluated using a best subset selection method with logistic regression. RESULTS Of the 63 included patients, 26 (41.3%) had CSF diversion at 6 months. Patients who required CSF diversion had a higher median FOHR (0.5 vs 0.4) and a higher percentage of postoperative intraventricular blood (30.8% vs 2.7%) compared with those who did not. A 0.1-unit increase in FOHR or intraventricular blood was associated with increased odds of CSF diversion (OR 2.9 [95% CI 1.3–7.8], p = 0.02 and OR 20.2 [95% CI 2.9–423.1], p = 0.01, respectively) with an overfitting-corrected concordance index of 0.68 (95% CI 0.56–0.80). CONCLUSIONS The preoperative FOHR and postoperative intraventricular blood were significant predictors of the need for permanent CSF diversion within 6 months after posterior fossa tumor resection in children.

In this paper, we propose {\it \underline{R}ecursive} {\it \underline{I}mportance} {\it \underline{S}ketching} algorithm for {\it \underline{R}ank} constrained least squares {\it \underline{O}ptimization} (RISRO). The key step of RISRO is recursive importance sketching, a new sketching framework based on deterministically designed recursive projections, which significantly differs from the randomized sketching in the literature \citep{mahoney2011randomized,woodruff2014sketching}. Several existing algorithms in the literature can be reinterpreted under this new sketching framework and RISRO offers clear advantages over them. RISRO is easy to implement and computationally efficient, where the core procedure in each iteration is to solve a dimension-reduced least squares problem. We establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. We also discover a deep connection of RISRO to the Riemannian Gauss-Newton algorithm on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO., Accepted by Operations Research

Tensor singular value decomposition (SVD) is a method to find a low-dimensional representation of data with meaningful structure in three or more dimensions. Tensor SVD has been applied to denoise atomic-resolution 4D scanning transmission electron microscopy (4D STEM) data. On data simulated from a SrTiO3 [100] perfect crystal and a Si [110] edge dislocation, tensor SVD achieved an average peak signal-to-noise ratio (PSNR) of ~40 dB, which matches or exceeds the performance of other denoising methods, with processing times at least 100 times shorter. On experimental data from SrTiO3 [100] and LiZnSb [11 2 ¯ 0]/GaSb [110] samples, tensor SVD denoises multiple GB 4D STEM data sets in ten minutes on a typical personal computer. Denoising with tensor SVD improves both convergent beam electron diffraction patterns and virtual-aperture annular dark field images.

The seminal work of Morgan and Rubin (2012) considers rerandomization for all the units at one time. In practice, however, experimenters may have to rerandomize units sequentially. For example, a clinician studying a rare disease may be unable to wait to perform an experiment until all the experimental units are recruited. Our work offers a mathematical framework for sequential rerandomization designs, where the experimental units are enrolled in groups. We formulate an adaptive rerandomization procedure for balancing treatment/control assignments over some continuous or binary covariates, using Mahalanobis distance as the imbalance measure. We prove in our key result, Theorem 3, that given the same number of rerandomizations (in expected value), under certain mild assumptions, sequential rerandomization achieves better covariate balance than rerandomization at one time., 23 pages

The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probability in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user-friendly schemes for developing non-asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which match the classic Hoeffding-type and Bernstein-type concentration inequalities, respectively. We also provide non-asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi-square, binomial, Poisson, Irwin-Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub-Gaussian and sub-exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally considered.

In this paper, we develop a novel procedure for low-rank tensor regression, namely \emph{\underline{I}mportance \underline{S}ketching \underline{L}ow-rank \underline{E}stimation for \underline{T}ensors} (ISLET). The central idea behind ISLET is \emph{importance sketching}, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical study that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods while having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$ orders of magnitude faster than baseline methods.

We propose a general semi-supervised inference framework focused on the estimation of the population mean. As usual in semi-supervised settings, there exists an unlabeled sample of covariate vectors and a labeled sample consisting of covariate vectors along with real-valued responses (“labels”). Otherwise, the formulation is “assumption-lean” in that no major conditions are imposed on the statistical or functional form of the data. We consider both the ideal semi-supervised setting where infinitely many unlabeled samples are available, as well as the ordinary semi-supervised setting in which only a finite number of unlabeled samples is available. ¶ Estimators are proposed along with corresponding confidence intervals for the population mean. Theoretical analysis on both the asymptotic distribution and $\ell_{2}$-risk for the proposed procedures are given. Surprisingly, the proposed estimators, based on a simple form of the least squares method, outperform the ordinary sample mean. The simple, transparent form of the estimator lends confidence to the perception that its asymptotic improvement over the ordinary sample mean also nearly holds even for moderate size samples. The method is further extended to a nonparametric setting, in which the oracle rate can be achieved asymptotically. The proposed estimators are further illustrated by simulation studies and a real data example involving estimation of the homeless population.

The completion of tensors, or high-order arrays, attracts significant attention in recent research. Current literature on tensor completion primarily focuses on recovery from a set of uniformly randomly measured entries, and the required number of measurements to achieve recovery is not guaranteed to be optimal. In addition, the implementation of some previous methods are NP-hard. In this article, we propose a framework for low-rank tensor completion via a novel tensor measurement scheme that we name Cross. The proposed procedure is efficient and easy to implement. In particular, we show that a third-order tensor of Tucker rank-$(r_{1},r_{2},r_{3})$ in $p_{1}$-by-$p_{2}$-by-$p_{3}$ dimensional space can be recovered from as few as $r_{1}r_{2}r_{3}+r_{1}(p_{1}-r_{1})+r_{2}(p_{2}-r_{2})+r_{3}(p_{3}-r_{3})$ noiseless measurements, which matches the sample complexity lower bound. In the case of noisy measurements, we also develop a theoretical upper bound and the matching minimax lower bound for recovery error over certain classes of low-rank tensors for the proposed procedure. The results can be further extended to fourth or higher-order tensors. Simulation studies show that the method performs well under a variety of settings. Finally, the procedure is illustrated through a real dataset in neuroimaging.

A key challenge in modeling single-cell RNA-seq (scRNA-seq) data is to capture the diverse gene expression states regulated by different transcriptional regulatory inputs across single cells, which is further complicated by a large number of observed zero and low expressions. We developed a left truncated mixture Gaussian (LTMG) model that stems from the kinetic relationships between the transcriptional regulatory inputs and metabolism of mRNA and gene expression abundance in a cell. LTMG infers the expression multi-modalities across single cell entities, representing a gene’s diverse expression states; meanwhile the dropouts and low expressions are treated as left truncated, specifically representing an expression state that is under suppression. We demonstrated that LTMG has significantly better goodness of fitting on an extensive number of single-cell data sets, comparing to three other state of the art models. In addition, our systems kinetic approach of handling the low and zero expressions and correctness of the identified multimodality are validated on several independent experimental data sets. Application on data of complex tissues demonstrated the capability of LTMG in extracting varied expression states specific to cell types or cell functions. Based on LTMG, a differential gene expression test and a co-regulation module identification method, namely LTMG-DGE and LTMG-GCR, are further developed. We experimentally validated that LTMG-DGE is equipped with higher sensitivity and specificity in detecting differentially expressed genes, compared with other five popular methods, and that LTMG-GCR is capable to retrieve the gene co-regulation modules corresponding to perturbed transcriptional regulations. A user-friendly R package with all the analysis power is available at https://github.com/zy26/LTMGSCA.

We developed a novel deconvolution method, namely Inference of Cell Types and Deconvolution (ICTD) that addresses the fundamental issue of identifiability and robustness in current tissue data deconvolution problem. ICTD provides substantially new capabilities for omics data based characterization of a tissue microenvironment, including (1) maximizing the resolution in identifying resident cell and sub types that truly exists in a tissue, (2) identifying the most reliable marker genes for each cell type, which are tissue and data set specific, (3) handling the stability problem with co-linear cell types, (4) co-deconvoluting with available matched multi-omics data, and (5) inferring functional variations specific to one or several cell types. ICTD is empowered by (i) rigorously derived mathematical conditions of identifiable cell type and cell type specific functions in tissue transcriptomics data and (ii) a semi supervised approach to maximize the knowledge transfer of cell type and functional marker genes identified in single cell or bulk cell data in the analysis of tissue data, and (iii) a novel unsupervised approach to minimize the bias brought by training data. Application of ICTD on real and single cell simulated tissue data validated that the method has consistently good performance for tissue data coming from different species, tissue microenvironments, and experimental platforms. Other than the new capabilities, ICTD outperformed other state-of-the-art devolution methods on prediction accuracy, the resolution of identifiable cell, detection of unknown sub cell types, and assessment of cell type specific functions. The premise of ICTD also lies in characterizing cell-cell interactions and discovering cell types and prognostic markers that are predictive of clinical outcomes.