The rapid growth of Internet of Things (IoT) devices necessitates efficient data compression techniques to handle the vast amounts of data generated by these devices. In this context, chemiresistive sensor arrays (CSAs), a simple-to-fabricate but crucial component in IoT systems, generate large volumes of data due to their simultaneous multi-sensor operations. Classical principal component analysis (cPCA) methods, a common solution to the data compression challenge, face limitations in preserving critical information during dimensionality reduction. In this study, we present quantum principal component analysis (qPCA) as a superior alternative to enhance information retention. Our findings demonstrate that qPCA outperforms cPCA in various back-end machine-learning modeling tasks, particularly in low-dimensional scenarios when limited Quantum bits (qubits) can be accessed. These results underscore the potential of noisy intermediate-scale quantum (NISQ) computers, despite current qubit limitations, to revolutionize data processing in real-world IoT applications, particularly in enhancing the efficiency and reliability of CSA data compression and readout.

Data exploration focuses on identifying informative patterns to discover new insight and knowledge about a collection of data. The often high-dimensional nature of such data renders the visual exploration process intractable for the human eye, and therefore specialized data manipulation of the original samples is essential in practice. Dimensionality reduction methods have been at the forefront of this challenge Bishop [2006] aiming to recover lower-dimensional embeddings of the original data that facilitate the identification of underlying data cohorts and help understand better the problem at hand. One of the most well known dimensionality reduction approaches perhaps is principal component analysis (PCA) Hotelling [1933], an efficient linear method aiming to maximizing the variance along the projection vectors, which in practice appears insufficient for meaningful separation of cohorts. A variety of non-linear methods have also been proposed that conversely focus on locally preserving the structure of the data such as Isomap Tenenbaum et al. [2000], LLE Roweis and Saul [2001], t-SNE van der Maaten and Hinton [2008], UMAP McInnes and Healy [2018], TriMap Amid and Warmuth [2019] and LargeVis Tang et al. [2016], etc. Projection pursuit (PP) Friedman and Tukey [1974], Caussinus and Ruiz-Gazen [2010] defines a family of dimensionality reduction methods that can enable various embedding effects depending on a suitably selected criterion. The kurtosis index Chiang et al. [2001] is one specific PP example that specializes in identifying "interesting" projections. Its minimization particularly penalizes the normality of the data distribution, promoting thus more meaningful separability when searching for clusters. The above approaches nevertheless share the same attribute of offering a single static projection that does not consider any prior knowledge a practitioner may have regarding the high-dimensional latent structure. Such projections can be uninformative as they tend to illustrate the most evident features which are often already known by the reader.

Finding informative low-dimensional representations that can be computed efficiently in large datasets is an important problem in data analysis. Recently, contrastive Principal Component Analysis (cPCA) was proposed as a more informative generalization of PCA that takes advantage of contrastive learning. However, the performance of cPCA is sensitive to hyper-parameter choice and there is currently no online algorithm for implementing cPCA. Here, we introduce a modified cPCA method, which we denote cPCA*, that is more interpretable and less sensitive to the choice of hyper-parameter. We derive an online algorithm for cPCA* and show that it maps onto a neural network with local learning rules, so it can potentially be implemented in energy efficient neuromorphic hardware. We evaluate the performance of our online algorithm on real datasets and highlight the differences and similarities with the original formulation.

The CP decomposition for high dimensional non-orthogonal spike tensors is an important problem with broad applications across many disciplines. However, previous works with theoretical guarantee typically assume restrictive incoherence conditions on the basis vectors for the CP components. In this paper, we propose new computationally efficient composite PCA and concurrent orthogonalization algorithms for tensor CP decomposition with theoretical guarantees under mild incoherence conditions. The composite PCA applies the principal component or singular value decompositions twice, first to a matrix unfolding of the tensor data to obtain singular vectors and then to the matrix folding of the singular vectors obtained in the first step. It can be used as an initialization for any iterative optimization schemes for the tensor CP decomposition. The concurrent orthogonalization algorithm iteratively estimates the basis vector in each mode of the tensor by simultaneously applying projections to the orthogonal complements of the spaces generated by others CP components in other modes. It is designed to improve the alternating least squares estimator and other forms of the high order orthogonal iteration for tensors with low or moderately high CP ranks. Our theoretical investigation provides estimation accuracy and statistical convergence rates for the two proposed algorithms. Our implementations on synthetic data demonstrate significant practical superiority of our approach over existing methods.

Principal component analysis (PCA; Jolliffe (2002)) is one of the most popular methods used to find a low-dimensional subspace in which a given dataset lies. Classical PCA (cPCA) can be formulated as a problem to find a subspace that minimizes the sum of squared residuals, but squared residuals make PCA vulnerable to outliers. A lot of PCA algorithms have been proposed to robustify cPCA. The R1-PCA proposed by Ding et al. (2006) replaced the sum of squared residuals in cPCA with the sum of unsquared ones. The optimal solution of R1-PCA has similar properties to those of cPCA, that is, it is given as the eigenvectors of the weighted covariance matrix and it is rotationally invariant. The absolute residuals can reduce negative impact of outliers, but an arbitrary large outlier can still break down the estimate. More recently, Zhang and Lerman (2014) and Lerman et al. (2015) relaxed the optimization problem so that the set of projection matrices is extended to a set of convex set of matrices, and 1

Contrastive learning (CL) is an emerging analysis approach that aims to discover unique patterns in one dataset relative to another. By applying this approach to network analysis, we can reveal unique characteristics in one network by contrasting with another. For example, with networks of protein interactions obtained from normal and cancer tissues, we can discover unique types of interactions in cancer tissues. However, existing CL methods cannot be directly applied to networks. To address this issue, we introduce a novel approach called contrastive network representation learning (cNRL). This approach embeds network nodes into a low-dimensional space that reveals the uniqueness of one network compared to another. Within this approach, we also design a method, named i-cNRL, that offers interpretability in the learned results, allowing for understanding which specific patterns are found in one network but not the other. We demonstrate the capability of i-cNRL with multiple network models and real-world datasets. Furthermore, we provide quantitative and qualitative comparisons across i-cNRL and other potential cNRL algorithm designs.

Dimensionality reduction (DR) is frequently used for analyzing and visualizing high-dimensional data as it provides a good first glance of the data. However, to interpret the DR result for gaining useful insights from the data, it would take additional analysis effort such as identifying clusters and understanding their characteristics. While there are many automatic methods (e.g., density-based clustering methods) to identify clusters, effective methods for understanding a cluster's characteristics are still lacking. A cluster can be mostly characterized by its distribution of feature values. Reviewing the original feature values is not a straightforward task when the number of features is large. To address this challenge, we present a visual analytics method that effectively highlights the essential features of a cluster in a DR result. To extract the essential features, we introduce an enhanced usage of contrastive principal component analysis (cPCA). Our method can calculate each feature's relative contribution to the contrast between one cluster and other clusters. With our cPCA-based method, we have created an interactive system including a scalable visualization of clusters' feature contributions. We demonstrate the effectiveness of our method and system with case studies using several publicly available datasets.

Principal component analysis (PCA) is widely used for feature extraction and dimensionality reduction, with documented merits in diverse tasks involving high-dimensional data. Standard PCA copes with one dataset at a time, but it is challenged when it comes to analyzing multiple datasets jointly. In certain data science settings however, one is often interested in extracting the most discriminative information from one dataset of particular interest (a.k.a. target data) relative to the other(s) (a.k.a. background data). To this end, this paper puts forth a novel approach, termed discriminative (d) PCA, for such discriminative analytics of multiple datasets. Under certain conditions, dPCA is proved to be least-squares optimal in recovering the component vector unique to the target data relative to background data. To account for nonlinear data correlations, (linear) dPCA models for one or multiple background datasets are generalized through kernel-based learning. Interestingly, all dPCA variants admit an analytical solution obtainable with a single (generalized) eigenvalue decomposition. Finally, corroborating dimensionality reduction tests using both synthetic and real datasets are provided to validate the effectiveness of the proposed methods.

We present a new technique called contrastive principal component analysis (cPCA) that is designed to discover low-dimensional structure that is unique to a dataset, or enriched in one dataset relative to other data. The technique is a generalization of standard PCA, for the setting where multiple datasets are available -- e.g. a treatment and a control group, or a mixed versus a homogeneous population -- and the goal is to explore patterns that are specific to one of the datasets. We conduct a wide variety of experiments in which cPCA identifies important dataset-specific patterns that are missed by PCA, demonstrating that it is useful for many applications: subgroup discovery, visualizing trends, feature selection, denoising, and data-dependent standardization. We provide geometrical interpretations of cPCA and show that it satisfies desirable theoretical guarantees. We also extend cPCA to nonlinear settings in the form of kernel cPCA. We have released our code as a python package and documentation is on Github.

Principal component analysis (PCA) has well-documented merits for data extraction and dimensionality reduction. PCA deals with a single dataset at a time, and it is challenged when it comes to analyzing multiple datasets. Yet in certain setups, one wishes to extract the most significant information of one dataset relative to other datasets. Specifically, the interest may be on identifying, namely extracting features that are specific to a single target dataset but not the others. This paper develops a novel approach for such so-termed discriminative data analysis, and establishes its optimality in the least-squares (LS) sense under suitable data modeling assumptions. The criterion reveals linear combinations of variables by maximizing the ratio of the variance of the target data to that of the remainders. The novel approach solves a generalized eigenvalue problem by performing SVD just once. Numerical tests using synthetic and real datasets showcase the merits of the proposed approach relative to its competing alternatives.