Despite the broad application of the analytic wavelet transform (AWT), a systematic statistical characterization of its magnitude and phase as inhomogeneous random fields on the time-frequency domain when the input is a random process remains underexplored. In this work, we study the magnitude and phase of the AWT as random fields on the time-frequency domain when the observed signal is a deterministic function plus additive stationary Gaussian noise. We derive their marginal and joint distributions, establish concentration inequalities that depend on the signal-to-noise ratio (SNR), and analyze their covariance structures. Based on these results, we derive an upper bound on the probability of incorrectly identifying the time-scale ridge of the clean signal, explore the regularity of scalogram contours, and study the relationship between AWT magnitude and phase. Our findings lay the groundwork for developing rigorous AWT-based algorithms in noisy environments.

Automated sleep stage classification is a valuable tool for analyzing sleep patterns and has numerous applications in wearable healthcare systems. However, the accuracy of sleep stage classification using signals from wearable devices can be affected by data quality issues such as signal interference or packet loss. In this study, we present an algorithm that addresses packet loss in respiratory inductive plethysmography (RIP) signals for sleep stage detection. RIP signals can be conveniently collected using abdominal and thoracic belts. By exploring the rich structural patterns in such signals, we utilize persistence diagrams to uncover macro-structures for sleep stage classification, which is particularly suitable for high data missing rates. Our model achieves a promising performance of 76% accuracy and a 0.54 Cohen’s kappa coefficient for three-stage classification. Additionally, we evaluate the model across different missing data rates and highlight the superior fault tolerance of persistence diagram features compared to other conventional temporal and spectral features.

Based on the Riemannian manifold model, we study the asymptotic behavior of a widely applied unsupervised learning algorithm, locally linear embedding (LLE), when the point cloud is sampled from a compact, smooth manifold with boundary. We show several peculiar behaviors of LLE near the boundary that are different from those diffusion-based algorithms. In particular, we show that LLE pointwisely converges to a mixed-type differential operator with degeneracy and we calculate the convergence rate. The impact of the hyperbolic part of the operator is discussed and we propose a clipped LLE algorithm which is a potential approach to recover the Dirichlet Laplace-Beltrami operator.

We systematically study the spectrum of kernel-based graph Laplacian (GL) constructed from high-dimensional and noisy random point cloud in the nonnull setup. The problem is motived by studying the model when the clean signal is sampled from a manifold that is embedded in a low-dimensional Euclidean subspace, and corrupted by high-dimensional noise. We quantify how the signal and noise interact in different regions of signal-to-noise ratio (SNR), and report the resulting peculiar spectral behavior of GL. In addition, we explore the impact of chosen kernel bandwidth on the spectrum of GL over different regions of SNR, which lead to an adaptive choice of kernel bandwidth that coincides with the common practice in real data. This result paves the way to a theoretical understanding of how practitioners apply GL when the dataset is noisy.

Motivated by analyzing long-term physiological time series, we design a robust and scalable spectral embedding algorithm that we refer to as RObust and Scalable Embedding via LANdmark Diffusion (Roseland). The key is designing a diffusion process on the dataset where the diffusion is done via a small subset called the landmark set. Roseland is theoretically justified under the manifold model, and its computational complexity is comparable with commonly applied subsampling scheme such as the Nyström extension. Specifically, when there are $n$ data points in $\mathbb{R}^q$ and $n^\beta$ points in the landmark set, where $\beta\in (0,1)$, the computational complexity of Roseland is $O(n^{1+2\beta}+qn^{1+\beta})$, while that of Nystrom is $O(n^{2.81\beta}+qn^{1+2\beta})$. To demonstrate the potential of Roseland, we apply it to { three} datasets and compare it with several other existing algorithms. First, we apply Roseland to the task of spectral clustering using the MNIST dataset (70,000 images), achieving 85\% accuracy when the dataset is clean and 78\% accuracy when the dataset is noisy. Compared with other subsampling schemes, overall Roseland achieves a better performance. Second, we apply Roseland to the task of image segmentation using images from COCO. Finally, we demonstrate how to apply Roseland to explore long-term arterial blood pressure waveform dynamics during a liver transplant operation lasting for 12 hours. In conclusion, Roseland is scalable and robust, and it has a potential for analyzing large datasets.

Diverse disciplines are interested in how the coordination of interacting agents' movements, emotions, and physiology over time impacts social behavior. Here, we describe a new multivariate procedure for automating the investigation of this kind of behaviorally-relevant “interactional synchrony”, and introduce a novel interactional synchrony measure based on features of dynamic time warping (DTW) paths. We demonstrate that our DTW path-based measure of interactional synchrony between facial action units of two people interacting freely in a natural social interaction can be used to predict how much trust they will display in a subsequent Trust Game. We also show that our approach outperforms univariate head movement models, models that consider participants' facial action units independently, and models that use previously proposed synchrony or similarity measures. The insights of this work can be applied to any research question that aims to quantify the temporal coordination of multiple signals over time, but has immediate applications in psychology, medicine, and robotics.

We apply local laws of random matrices and free probability theory to study the spectral properties of two kernel-based sensor fusion algorithms, nonparametric canonical correlation analysis (NCCA) and alternating diffusion (AD), for two simultaneously recorded high dimensional datasets under the null hypothesis. The matrix of interest is the product of the kernel matrices associated with the databsets, which may not be diagonalizable in general. We prove that in the regime where dimensions of both random vectors are comparable to the sample size, if NCCA and AD are conducted using a smooth kernel function, then the first few nontrivial eigenvalues will converge to real deterministic values provided the datasets are independent Gaussian random vectors. Toward the claimed result, we also provide a convergence rate of eigenvalues of a kernel affinity matrix.

We propose a novel mathematical framework to address the problem of automatically solving large jigsaw puzzles. This problem assumes a large image, which is cut into equal square pieces that are arbitrarily rotated and shuffled, and asks to recover the original image given the transformed pieces. The main contribution of this work is a method for recovering the rotations of the pieces when both shuffles and rotations are unknown. A major challenge of this procedure is estimating the graph connection Laplacian without the knowledge of shuffles. A careful combination of our proposed method for estimating rotations with any existing method for estimating shuffles results in a practical solution for the jigsaw puzzle problem. Our theory guarantees, in a clean setting, that our basic idea of recovering rotations is robust to some corruption of the connection graph. Numerical experiments demonstrate the competitive accuracy of this solution, its robustness to corruption, and its computational advantage for large puzzles.

Extracting the instantaneous heart rate (iHR) from face videos has been well studied in recent years. It is well known that changes in skin color due to blood flow can be captured using conventional cameras. One of the main limitations of methods that rely on this principle is the need of an illumination source. Moreover, they have to be able to operate under different light conditions. One way to avoid these constraints is using infrared cameras, allowing the monitoring of iHR under low light conditions. In this work, we present a simple, principled signal extraction method that recovers the iHR from infrared face videos. We tested the procedure on 7 participants, for whom we recorded an electrocardiogram simultaneously with their infrared face video. We checked that the recovered signal matched the ground truth iHR, showing that infrared is a promising alternative to conventional video imaging for heart rate monitoring, especially in low light conditions. Code is available at https://github.com/natalialmg/IR_iHR.

Kernel mixture models are routinely used for density estimation. However, in multivariate settings, issues arise in efficiently approximating lower-dimensional structure in the data. For example, it is common to suppose that the density is concentrated near a lower-dimensional non-linear subspace or manifold. Typical kernels used to locally approximate such subspaces are inflexible, so that a large number of components are often needed. We propose a novel class of LOcally COnvex (LOCO) kernels that are flexible in adapting to nonlinear local structure. LOCO kernels are induced by introducing random knots within local neighborhoods, and generating data as a random convex combination of these knots with adaptive weights and an additive noise. For identifiability, we constrain all observations from a particular component to have the same mean. For Bayesian inference subject to this constraint, we develop a hybrid Gibbs sampler and optimization algorithm that incorporates a Lagrange multiplier within a splitting method. The resulting LOCO algorithm is shown to dramatically outperform typical Gaussian mixture models in challenging examples.

This paper presents a novel signal compression algorithm based on the Blaschke unwinding adaptive Fourier decomposition (AFD). The Blaschke unwinding AFD is a newly developed signal decomposition theory. It utilizes the Nevanlinna factorization and the maximal selection principle in each decomposition step, and achieves a faster convergence rate with higher fidelity. The proposed compression algorithm is applied to the electrocardiogram signal. To assess the performance of the proposed compression algorithm, in addition to the generic assessment criteria, we consider the less discussed criteria related to the clinical needs-for the heart rate variability analysis purpose, how accurate the R-peak information is preserved is evaluated. The experiments are conducted on the MIT-BIH arrhythmia benchmark database. The results show that the proposed algorithm performs better than other state-of-the-art approaches. Meanwhile, it also well preserves the R-peak information.

To meet the demands on a comfortable screening, or even diagnostic, equipment without interfering with the sleep, this study develops a level IV portable system, equipped with two tri-axial accelerometers (TAA) measuring the thoracic and abdominal respiratory efforts, and one oximeter measuring the oxygen saturation (SpO2), to identify obstructive sleep apnea (OSA), central sleep apnea (CSA), and hypopnea (HYP) events. The prototype integrates all the hardware and software for physiological information extraction. In addition, an automatic event detection algorithm is proposed to reduce the labor-intensive work on scoring the events. Based on 63 subjects, with 80% data for training and 20% for validation, the classification accuracy of the apnea hypopnea-index (AHI) is 84.13%. The results indicate that the proposed algorithm has great potential to classify the severity of patients in clinical examinations for both the screening and the homecare purposes.

Physiologically, the thoracic (THO) and abdominal (ABD) movement signals, captured using wearable piezo-electric bands, provide information about various types of apnea, including central sleep apnea (CSA) and obstructive sleep apnea (OSA). However, the use of piezo-electric wearables in detecting sleep apnea events has been seldom explored in the literature. This study explored the possibility of identifying sleep apnea events, including OSA and CSA, by solely analyzing one or both the THO and ABD signals. An adaptive non-harmonic model was introduced to model the THO and ABD signals, which allows us to design features for sleep apnea events. To confirm the suitability of the extracted features, a support vector machine was applied to classify three categories - normal and hypopnea, OSA, and CSA. According to a database of 34 subjects, the overall classification accuracies were on average 75.9%±11.7% and 73.8%±4.4%, respectively, based on the cross validation. When the features determined from the THO and ABD signals were combined, the overall classification accuracy became 81.8%±9.4%. These features were applied for designing a state machine for online apnea event detection. Two event-byevent accuracy indices, S and I, were proposed for evaluating the performance of the state machine. For the same database, the S index was 84.01%±9.06%, and the I index was 77.21%±19.01%. The results indicate the considerable potential of applying the proposed algorithm to clinical examinations for both screening and homecare purposes.

In this paper, we address the problem of multimodal signal processing and present a manifold learning method to extract the common source of variability from multiple measurements. This method is based on alternating-diffusion and is particularly adapted to time series. We show that the common source of variability is extracted from multiple sensors as if it were the only source of variability, extracted by a standard manifold learning method from a single sensor, without the influence of the sensor-specific variables. In addition, we present application to sleep stage assessment. We demonstrate that, indeed, through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.

Computerized tomography is a standard method for obtaining internal structure of objects from their projection images. While CT reconstruction requires the knowledge of the imaging directions, there are some situations in which the imaging directions are unknown, for example, when imaging a moving object. It is therefore desirable to design a reconstruction method from projection images taken at unknown directions. Another difficulty arises from the fact that the projections are often contaminated by noise, practically limiting all current methods, including the recently proposed diffusion map approach. In this paper, we introduce two denoising steps that allow reconstructions at much lower signal-to-noise ratios (SNRs) when combined with the diffusion map framework. In the first denoising step we use principal component analysis (PCA) together with classical Wiener filtering to derive an asymptotically optimal linear filter. In the second step, we denoise the graph of similarities between the filtered projections using a network analysis measure such as the Jaccard index. Using this combination of PCA, Wiener filtering, graph denoising, and diffusion maps, we are able to reconstruct the two-dimensional (2-D) Shepp-Logan phantom from simulative noisy projections at SNRs well below their currently reported threshold values. We also report the results of a numerical experiment corresponding to an abdominal CT. Although the focus of this paper is the 2-D CT reconstruction problem, we believe that the combination of PCA, Wiener filtering, graph denoising, and diffusion maps is potentially useful in other signal processing and image analysis applications.