Sparse principal component analysis (SPCA) is widely used for dimensionality reduction and feature extraction in high-dimensional data analysis. Despite many methodological and theoretical developments in the past two decades, the theoretical guarantees of the popular SPCA algorithm proposed by [1] based on the elastic net are still unknown. This paper aims to address this critical theoretical gap. We first revisit the SPCA algorithm of [1] and present our implementation. We also study a computationally more efficient variant of the SPCA algorithm in [1] that can be considered as the limiting case of SPCA. We provide the guarantees of convergence to a stationary point for both algorithms and prove that, under a sparse spiked covariance model, both algorithms can recover the principal subspace consistently under mild regularity conditions. We show that their estimation error bounds match the best available bounds of existing works or the minimax rates up to some logarithmic factors. Moreover, we demonstrate the competitive numerical performance of both algorithms in numerical experiments.

We present the subspace-constrained Tyler's estimator (STE) designed for recovering a low-dimensional subspace within a dataset that may be highly corrupted with outliers. STE is a fusion of the Tyler's M-estimator (TME) and a variant of the fast median subspace. Our theoretical analysis suggests that, under a common inlier-outlier model, STE can effectively recover the underlying subspace, even when it contains a smaller fraction of inliers relative to other methods in the field of robust subspace recovery. We apply STE in the context of Structure from Motion (SfM) in two ways: for robust estimation of the fundamental matrix and for the removal of outlying cameras, enhancing the robustness of the SfM pipeline. Numerical experiments confirm the state-of-the-art performance of our method in these applications. This research makes significant contributions to the field of robust subspace recovery, particularly in the context of computer vision and 3D reconstruction.

The paper considers a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. Jing et al. (2020) introduced the MMLSBM and developed a clustering methodology, TWIST, based on regularized tensor decomposition. The present paper proposes a different technique, an alternating minimization algorithm (ALMA), that aims at simultaneous recovery of the layer partition, together with estimation of the matrices of connection probabilities of the distinct layers. Compared to TWIST, ALMA achieves higher accuracy, both theoretically and numerically.

Platooning of connected vehicles is a solution geared toward improving traffic throughput, highway safety, driving comfort, and fuel efficiency. These vehicles are equipped with Cooperative Adaptive Cruise Controller (CACC) that integrates information from dedicated short-range communication (DSRC) radio and sensors for safe navigation. The possibility of malicious attacks such as Denial of Service (DoS) or False Data Injection (FDI) on sensor data or control inputs tends to affect reliability, and jeopardize the safety of connected vehicles. Thus, securing sensor data of these vehicles from DoS or FDI attacks is essential to avoid unwanted consequences. To withstand sensor attacks, resilient state estimators have been developed for networked cyber-physical systems (CPS). However, such estimators do not perform well as the number of compromised sensors of the system increases. As such, we propose a novel convex optimization based Resilient Distributed State Estimator (RDSE) that bounds the state estimation error, irrespective of the magnitude of the attack and the number of compromised sensors. We theoretically prove that the proposed estimator has similar performance compared to the state-of-the-art Distributed Kalman Filter (DKF) under attack free and noise free scenarios. While under attack, our RDSE outperforms the DKF and we provide a theoretical bound on state estimation error generated by RDSE during an attack. We also demonstrate the effectiveness of RDSE against FDI attacks in a platoon with five vehicles and compare its performance during attack against the DKF and the Resilient Distributed Kalman Filter (RDKF).

In this paper, we study the collaboratively estimating problem of a discrete-time LTI system with a time-varying undirected communication graph among sensors. The performance of resilient state estimators developed for cyber-physical systems (CPS) degenerates if the sensor measurements are compromised. To obtain robustness for the estimation, we propose Resilient Distributed Estimator with Information Consensus (RDEIC). RDEIC is a consensus-based resilient distributed algorithm that produces bounded state estimation errors with faulty sensors. Our algorithm converges to the true state in an attack-free scenario and it produces bounded estimation errors during an attack. The performance of the proposed algorithm is demonstrated with Matlab simulations.

MOTIVATION: Accurate estimation of transcript isoform abundance is critical for downstream transcriptome analyses and can lead to precise molecular mechanisms for understanding complex human diseases, like cancer. Simplex mRNA Sequencing (RNA-Seq) based isoform quantification approaches are facing the challenges of inherent sampling bias and unidentifiable read origins. A large-scale experiment shows that the consistency between RNA-Seq and other mRNA quantification platforms is relatively low at the isoform level compared to the gene level. In this project, we developed a platform-integrated model for transcript quantification (IntMTQ) to improve the performance of RNA-Seq on isoform expression estimation. IntMTQ, which benefits from the mRNA expressions reported by the other platforms, provides more precise RNA-Seq-based isoform quantification and leads to more accurate molecular signatures for disease phenotype prediction. RESULTS: In the experiments to assess the quality of isoform expression estimated by IntMTQ, we designed three tasks for clustering and classification of 46 cancer cell lines with four different mRNA quantification platforms, including newly developed NanoString's nCounter technology. The results demonstrate that the isoform expressions learned by IntMTQ consistently provide more and better molecular features for downstream analyses compared with five baseline algorithms which consider RNA-Seq data only. An independent RT-qPCR experiment on seven genes in twelve cancer cell lines showed that the IntMTQ improved overall transcript quantification. The platform-integrated algorithms could be applied to large-scale cancer studies, such as The Cancer Genome Atlas (TCGA), with both RNA-Seq and array-based platforms available. AVAILABILITY AND IMPLEMENTATION: Source code is available at: https://github.com/CompbioLabUcf/IntMTQ. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Detecting cancer gene expression and transcriptome changes with mRNA-sequencing or array-based data are important for understanding the molecular mechanisms underlying carcinogenesis and cellular events during cancer progression. In previous studies, the differentially expressed genes were detected across patients in one cancer type. These studies ignored the role of mRNA expression changes in driving tumorigenic mechanisms that are either universal or specific in different tumor types. To address the problem, we introduce two network-based multi-task learning frameworks, NetML and NetSML, to discover common differentially expressed genes shared across different cancer types as well as differentially expressed genes specific to each cancer type. The proposed frameworks consider the common latent gene co-expression modules and gene-sample biclusters underlying the multiple cancer datasets to learn the knowledge crossing different tumor types. RESULTS: Large-scale experiments on simulations and real cancer high-throughput datasets validate that the proposed network-based multi-task learning frameworks perform better sample classification compared with the models without the knowledge sharing across different cancer types. The common and cancer-specific molecular signatures detected by multi-task learning frameworks on The Cancer Genome Atlas ovarian, breast and prostate cancer datasets are correlated with the known marker genes and enriched in cancer-relevant Kyoto Encyclopedia of Genes and Genome pathways and gene ontology terms. AVAILABILITY AND IMPLEMENTATION: Source code is available at: https://github.com/compbiolabucf/NetML. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

The performance of resilient state estimators developed for cyber-physical systems (CPSs) decreases as the number of compromised sensors of the system increases. Furthermore, some of these algorithms leverage computationally expensive optimization techniques to incorporate resiliency. As such, we propose a fast resilient distributed state estimator (FRDSE), which is a novel resilient distributed algorithm that produces bounded state estimation errors regardless of the magnitude of the attack and the number of compromised sensors. Our algorithm converges to the true state in an attack-free and noise-free scenario and it produces bounded estimation errors during an attack. Compared to existing algorithms, FRDSE is more computationally efficient. We provide theoretical guarantees on the convergence of FRDSE in attack-free scenario and prove its resiliency during an attack. We demonstrate the performance of our algorithm against false data injection (FDI) attack in a platoon of vehicles and compare its runtime against existing algorithms. We observe that on a platoon of eight vehicles, runtime of our algorithm is 0.102 s, much lower than the state-of-the-art solutions.

We consider the phase retrieval problem, where the goal is to reconstruct an n-dimensional complex vector from its phaseless scalar products with m sensing vectors, independently sampled from complex normal distributions. We show that, if m ≥ Mn3/2log7/2n for some M > 0, then the classical algorithm of alternating minimization with random initialization succeeds with high probability as n, m → ∞. This is a step toward proving the conjecture in, which conjectures that the algorithm succeeds when m = O(n). The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.

We present a mathematical analysis of a non-convex energy landscape for robust subspace recovery. We prove that an underlying subspace is the only stationary point and local minimizer in a specified neighborhood under a deterministic condition on a dataset. If the deterministic condition is satisfied, we further show that a geodesic gradient descent method over the Grassmannian manifold can exactly recover the underlying subspace when the method is properly initialized. Proper initialization by principal component analysis is guaranteed with a simple deterministic condition. Under slightly stronger assumptions, the gradient descent method with a piecewise constant step-size scheme achieves linear convergence. The practicality of the deterministic condition is demonstrated on some statistical models of data, and the method achieves almost state-of-the-art recovery guarantees on the Haystack Model for different regimes of sample size and ambient dimension. In particular, when the ambient dimension is fixed and the sample size is large enough, we show that our gradient method can exactly recover the underlying subspace for any fixed fraction of outliers (less than 1).

Automotive systems have always been designed with safety in mind. In this regard, the functional safety standard, ISO 26262, was drafted with the intention of minimizing risk due to random hardware faults or systematic failure in design of electrical and electronic components of an automobile. However, growing complexity of a modern car has added another potential point of failure in the form of cyber or sensor attacks. Recently, researchers have demonstrated that vulnerability in vehicle's software or sensing units could enable them to remotely alter the intended operation of the vehicle. As such, in addition to safety, security should be considered as an important design goal. However, designing security solutions without the consideration of safety objectives could result in potential hazards. Consequently, in this paper we propose the notion of security for safety and show that by integrating safety conditions with our system-level security solution, which comprises of a modified Kalman filter and a Chi-squared detector, we can prevent potential hazards that could occur due to violation of safety objectives during an attack. Furthermore, with the help of a car-following case study, where the follower car is equipped with an adaptive-cruise control unit, we show that our proposed system-level security solution preserves the safety constraints and prevent collision between vehicle while under sensor attack.

We establish exact recovery for the Least Unsquared Deviations (LUD) algorithm of Özyeşil and Singer. More precisely, we show that for sufficiently many cameras with given corrupted pairwise directions, where both camera locations and pairwise directions are generated by a special probabilistic model, the LUD algorithm exactly recovers the camera locations with high probability. A similar exact recovery guarantee for camera locations was established for the ShapeFit algorithm by Hand, Lee, and Voroninski, but with typically less corruption.

The introduction of automation in cyber-physical systems (CPS) has raised major safety and security concerns. One attack vector is the sensing unit whose measurements can be manipulated by an adversary through attacks such as denial of service and delay injection. To secure an autonomous CPS from such attacks, we use a challenge response authentication (CRA) technique for detection of attack in active sensors data and estimate safe measurements using the recursive least square algorithm. For demonstrating effectiveness of our proposed approach, a car-follower model is considered where the follower vehicle's radar sensor measurements are manipulated in an attempt to cause a collision.

We propose a spectral clustering method based on local principal components analysis (PCA). After performing local PCA in selected neighborhoods, the algorithm builds a nearest neighbor graph weighted according to a discrepancy between the principal subspaces in the neighborhoods, and then applies spectral clustering. As opposed to standard spectral methods based solely on pairwise distances between points, our algorithm is able to resolve intersections. We establish theoretical guarantees for simpler variants within a prototypical mathematical framework for multi-manifold clustering, and evaluate our algorithm on various simulated data sets.

We consider the problem of finding lower dimensional subspaces in the presence of outliers and noise in the online setting. In particular, we extend previous batch formulations of robust PCA to the stochastic setting with minimal storage requirements and runtime complexity. We introduce three novel stochastic approximation algorithms for robust PCA that are extensions of standard algorithms for PCA - the stochastic power method, incremental PCA and online PCA using matrix-exponentiated-gradient (MEG) updates. For robust online PCA we also give a sub-linear convergence guarantee. Our numerical results demonstrate the superiority of the the robust online method over the other robust stochastic methods and the advantage of robust methods over their non-robust counterparts in the presence of outliers in artificial and real scenarios.

We present a simple and fast geometric method for modeling data by a union of affine subspaces. The method begins by forming a collection of local best-fit affine subspaces, i.e., subspaces approximating the data in local neighborhoods. The correct sizes of the local neighborhoods are determined automatically by the Jones’ β 2 numbers (we prove under certain geometric conditions that our method finds the optimal local neighborhoods). The collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model. We discuss applications to tracking-based motion segmentation and clustering of faces under different illuminating conditions. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the MNIST handwritten digits data; and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces.

The hybrid linear modeling problem is to identify a set of d-dimensional affine sets in RD. It arises, for example, in object tracking and structure from motion. The hybrid linear model can be considered as the second simplest (behind linear) manifold model of data. In this paper we will present a very simple geometric method for hybrid linear modeling based on selecting a set of local best fit flats that minimize a global ℓ1error measure. The size of the local neighborhoods is determined automatically by the Jones' β2numbers; it is proven under certain geometric conditions that good local neighborhoods exist and are found by our method. We also demonstrate how to use this algorithm for fast determination of the number of affine subspaces. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the algorithm on synthetic and real hybrid linear data.