arXiv subjects

Hau-Tieng Wu

Publications and source records attributed to Hau-Tieng Wu.

At least 19 recordsLinked to original sources

arXiv2026

Time-Varying Spiky Wave-Shape Functions for Non-Stationary Signal Decomposition

We propose a novel framework for decomposing nonstationary spiky signals. Unlike classical adaptive non-harmonic models, which represent signals as amplitude- and frequency-modulated (AM--FM) oscillations, the proposed model is designed for signals whose dominant structures are highly localized, impulsive, or spike-like, and contains physiological variability, which is challenging to be modeled as AM--FM representations. Representative examples include electrocardiogram (ECG) complexes, epileptic electroencephalogram (EEG) transients, and other pulse-like physiological signals. We first introduce a fixed-waveform model for repetitive spiky structures and then extend it to accommodate cycle-to-cycle waveform variability. The resulting algorithm, termed \emph{Spiky Shape-adaptive Mode Decomposition} (SSAMD), estimates waveforms in the Fourier coefficient domain. Fourier coefficients are first estimated via local regression and then regularized by exploiting their low-dimensional manifold structure. Specifically, we combine a PCA-based parametrization with an entropy regularization acting on the singular-value spectrum of the coefficient matrix, promoting morphological consistency while preserving structured waveform variability. The proposed framework is flexible and overcomes the limitations of existing models. We validate the method on synthetic and real biomedical signals, including ECG with atrial fibrillation, epileptic EEG, and trans-abdominal maternal ECG. Experimental results demonstrate effective denoising, waveform tracking, decomposition, and segmentation, while preserving physiologically meaningful morphological variations.

eess.SP↗

arXiv2026

Data-driven time-frequency tessellation for signals with oscillatory amplitude envelopes and instantaneous frequency, with application to photoplethysmograhy

Biomedical signals often comprise multiple non-sinusoidal oscillatory components whose amplitude modulation (AM) and instantaneous frequency (IF) may themselves be governed by additional (second-order) oscillatory dynamics with time-varying amplitude and frequency. We introduce a novel time-frequency (TF) analysis framework, {\em Tessellation-based Ensembled Time-Frequency Representation via Integrated Shifting} (TETRIS), designed based on the proposed generalized adaptive non-harmonic model to leverage second-order oscillatory information in this class of signals. We present the model and algorithm using the photoplethysmogram (PPG) as a canonical example, whose cardiac component is known to encode respiratory information in both AM and IF, and demonstrate how respiratory signals can be recovered from PPG. The central idea of TETRIS is to partition the TF plane along the estimated IF of the cardiac component and to process each partition adaptively to enhance representation quality. This tessellation enables a refined time-frequency representation (TFR), allowing more effective recovery of the respiratory modulation governing the AM of the cardiac component. We provide theoretical justification for the proposed method and validate its performance on semi-synthetic signals. Finally, we demonstrate that TETRIS enables improved reconstruction of multiple surrogate respiratory signals directly from PPG data. While the model and algorithm are developed with a focus on PPG, the framework is flexible and has potential to be applied to other signals.

stat.AP↗

arXiv2026

Fingertip Micro-Motion as a Source of Respiratory Information During Sleep Using Triaxial Accelerometers

Objective: Triaxial accelerometers (TAAs) are widely used in homecare medicine. This study investigates whether TAA signals recorded at the fingertip encode respiratory information, particularly instantaneous respiratory rate (IRR) and respiratory effort, during sleep. Method: We propose an antiderivative-based nonlinear transformation to convert TAA signals into a respiratory surrogate, termed TAA-resp. To quantify the embedded respiratory-induced motion, a modern time-frequency analysis tool is applied to derive an index, referred to as the respiratory motion index (RMI). The proposed TAA-resp and RMI are validated on a dataset comprising 39 full-night recordings with simultaneous polysomnography (PSG) and a fingertip TAA measurements. Criteria for labeling TAA-resp signal quality as good, moderate, or poor are established, and expert annotations are obtained. Result: On average, TAA-resp over 22.2% $\pm$ 15.6% of full-night recordings encodes high-quality respiratory information, reaching up to 58.9% in some cases. TAA-resp shows stronger correlation with thoracic and abdominal motion than with airflow, indicating predominant capture of respiratory effort. High-quality TAA-resp offers an accurate IRR estimate with root mean square error $0.027 \pm 0.022$ Hz. RMI is higher for high-quality segments and lower for poor-quality segments, and its distribution aligns with physiology, with higher values during REM, N2, and N3 sleep and in the absence of apnea or hypopnea events. In leave-one-subject-out cross-validation, RMI predicts quality labels with 0.74 sensitivity and 0.75 specificity. Conclusion: Fingertip-mounted TAAs encode meaningful respiratory information. Leveraging this underutilized signal may enhance home-based sleep monitoring in channel-limited settings.

physics.med-ph↗

arXiv2026

Accelerate Vector Diffusion Maps by Landmarks

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.

stat.ML↗

arXiv2026

Functional Estimation of Manifold-Valued Diffusion Processes

Nonstationary high-dimensional time series are increasingly encountered in biomedical research as measurement technologies advance. Owing to the homeostatic nature of physiological systems, such datasets are often located on, or can be well approximated by, a low-dimensional manifold. Modeling such datasets by manifold-valued It\^o diffusion processes has been shown to provide valuable insights and to guide the design of algorithms for clinical applications. In this paper, we propose Nadaraya-Watson type nonparametric estimators for the drift vector field and diffusion matrix of the process from one trajectory. Assuming a time-homogeneous stochastic differential equation on a smooth complete manifold without boundary, we show that as the sampling interval and kernel bandwidth vanish with increasing trajectory length, recurrence of the process yields asymptotic consistency and normality of the drift and diffusion estimators, as well as the associated occupation density. Analysis of the diffusion estimator further produces a tangent space estimator for dependent data, which has its own interest and is essential for drift estimation. Numerical experiments across a range of manifold configurations support the theoretical results.

stat.ME↗

arXiv2026

Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noise

Multiview datasets are common in scientific and engineering applications, yet existing fusion methods offer limited theoretical guarantees, particularly in the presence of heterogeneous and high-dimensional noise. We propose Generalized Robust Adaptive-Bandwidth Multiview Diffusion Maps (GRAB-MDM), a new kernel-based diffusion geometry framework for integrating multiple noisy data sources. The key innovation of GRAB-MDM is a {view}-dependent bandwidth selection strategy that adapts to the geometry and noise level of each view, enabling a stable and principled construction of multiview diffusion operators. Under a common-manifold model, we establish asymptotic convergence results and show that the adaptive bandwidths lead to provably robust recovery of the shared intrinsic structure, even when noise levels and sensor dimensions differ across views. Numerical experiments demonstrate that GRAB-MDM significantly improves robustness and embedding quality compared with fixed-bandwidth and equal-bandwidth baselines, and usually outperform existing algorithms. The proposed framework offers a practical and theoretically grounded solution for multiview sensor fusion in high-dimensional noisy environments.

stat.ML↗

arXiv2026

On spectral interference of the short-time Fourier transform and its nonlinear variations

Spectral interference, the frequency counterpart of the beating phenomenon in the time domain, can severely distort time-frequency representations (TFRs) in physical applications. We study this phenomenon for the short-time Fourier transform (STFT) with a Gaussian window and for nonlinear refinements based on the reassignment method, with an emphasis on the synchrosqueezing transform (SST). Working with a two-component harmonic model, we quantify when STFT can (and cannot) resolve two nearby frequencies: a sharp transition occurs at a critical gap that scales inversely to kernel bandwidth and depends explicitly on the amplitude ratio. Below this threshold, the spectrogram ridges undergo bifurcation and form repeating time-frequency bubbles, which we describe asymptotically and, in the balanced-amplitude case, approximate closely by ellipses. We then analyze the STFT phase, showing a canonical winding behavior, and relate the complex-valued SST reassignment map to a holomorphic structure via the Bargmann transform. In the two-component setting the reassignment rule admits an explicit Mobius-geometry description, sending frequency lines to circular arcs in the complex plane. Finally, viewing SST and reassignment through a measure mapping perspective, we derive small-kernel asymptotics that explain when reassignment sharpens energy and when it produces distorted or misleading TFRs; we also introduce a generalized synchrosqueezing framework that isolates the role of STFT weighting and clarifies how alternative choices can mitigate interference in certain regimes.

math.CA↗

arXiv2025

Theoretical analysis of phase-rectified signal averaging (PRSA) algorithm

Phase-rectified signal averaging (PRSA) is a widely used algorithm to analyze nonstationary biomedical time series. The method operates by identifying hinge points in the time series according to prescribed rules, extracting segments centered at these points (with overlap permitted), and then averaging the segments. The resulting output is intended to capture the underlying quasi-oscillatory pattern of the signal, which can subsequently serve as input for further scientific analysis. However, a theoretical analysis of PRSA is lacking. In this paper, we investigate PRSA under two settings. First, when the input consists of a superposition of two oscillatory components, $\cos(2\pi t)+A\cos(2\pi (\xi t+\phi))$, where $A>0$, $\xi\in (0,1)$ and $\phi\in [0,1)$, we show that, asymptotically when the sample size $n\to \infty$, the PRSA output takes the form $A'\sin(2\pi t)+B'\sin(2\pi \xi t)$, where $A',B'\neq 0$. Second, when the input is a stationary Gaussian random process, we establish a central limit theorem: under mild regularity conditions, the averaged vector produced by PRSA converges in distribution to a Gaussian random vector as $n\to \infty$ with mean determined by the covariance structure of the random process. These results indicate that caution is warranted when interpreting PRSA outputs for scientific applications.

math.ST↗

arXiv2025

Time-Frequency Filtering Meets Graph Clustering

We show that the problem of identifying different signal components from a time-frequency representation can be equivalently phrased as a graph clustering problem: given a graph $G=(V,E)$ one aims to identify `clusters', subgraphs that are strongly connected and have relatively few connections between them. The graph clustering problem is well studied, we show how these ideas can suggest (many) new ways to identify signal components. Numerical experiments illustrate the ideas.

eess.SP↗

arXiv2025

Efficient Artifacts Removal for Adaptive Deep Brain Stimulation and a Temporal Event Localization Analysis

Adaptive deep brain stimulation (aDBS) leverages symptom-related biomarkers to deliver personalized neuromodulation therapy, with the potential to improve treatment efficacy and reduce power consumption compared to conventional DBS. However, stimulation-induced signal contamination remains a major technical barrier to advancing its clinical application. Existing artifact removal strategies, both front-end and back-end, face trade-offs between artifact suppression and algorithmic flexibility. Among back-end algorithms, Shrinkage and Manifold-based Artifact Removal using Template Adaptation (SMARTA) has shown promising performance in mitigating stimulus artifacts with minimal distortion to local field potentials (LFPs), but its high computational demand and inability to handle transient direct current (DC) artifacts limit its use in real-time applications. To address this, we developed SMARTA+, a computationally efficient extension of SMARTA capable of suppressing both stimulus and transient DC artifacts while supporting flexible algorithmic design. We evaluated SMARTA+ using semi-real aDBS data and real data from Parkinson's disease patients. Compared to SMARTA and other established methods, SMARTA+ achieved comparable or superior artifact removal while significantly reducing computation time. It preserved spectral and temporal structures, ranging from beta band to high-frequency oscillations, and demonstrated robustness across diverse stimulation protocols. Temporal event localization analysis further showed improved accuracy in detecting beta bursts. These findings support SMARTA+ as a promising tool for advancing real-time, closed-loop aDBS systems.

eess.SP↗

arXiv2025

On Random Fields Associated with Analytic Wavelet Transform

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.

math.ST↗

arXiv2025

On the Practical Use of Blaschke Decomposition in Nonstationary Signal Analysis

The Blaschke decomposition-based algorithm, {\em Phase Dynamics Unwinding} (PDU), possesses several attractive theoretical properties, including fast convergence, effective decomposition, and multiscale analysis. However, its application to real-world signal decomposition tasks encounters notable challenges. In this work, we propose two techniques, divide-and-conquer via tapering and cumulative summation (cumsum), to handle complex trends and amplitude modulations and the mode-mixing caused by winding. The resulting method, termed {\em windowed PDU}, enhances PDU's performance in practical decomposition tasks. We validate our approach through both simulated and real-world signals, demonstrating its effectiveness across diverse scenarios.

stat.ME↗

arXiv2025

Uncertainty quantification of synchrosqueezing transform under complicated nonstationary noise

We propose a bootstrapping framework to quantify uncertainty in time-frequency representations (TFRs) generated by the short-time Fourier transform (STFT) and the STFT-based synchrosqueezing transform (SST) for oscillatory signals with time-varying amplitude and frequency contaminated by complex nonstationary noise. To this end, we leverage a recent high-dimensional Gaussian approximation technique to establish a sequential Gaussian approximation for nonstationary processes under mild assumptions. This result is of independent interest and provides a theoretical basis for characterizing the approximate Gaussianity of STFT-induced TFRs as random fields. Building on this foundation, we establish the robustness of SST-based signal decomposition in the presence of nonstationary noise. Furthermore, assuming locally stationary noise, we develop a Gaussian autoregressive bootstrap for uncertainty quantification of SST-based TFRs and provide theoretical justification. We validate the proposed methods with simulations and illustrate their practical utility by analyzing spindle activity in electroencephalogram recordings. Our work bridges time-frequency analysis in signal processing and nonlinear spectral analysis of time series in statistics.

stat.ME↗

arXiv2024

Probabilistic Analysis of Scalogram Ridges in Signal Processing

While ridges in the scalogram, determined by the squared modulus of analytic wavelet transform (AWT), is a widely accepted concept and utilized in nonstationary time series analysis, their behavior in noisy environments remains underexplored. Our object is to provide a theoretical foundation for scalogram ridges by defining ridges as a potentially set-valued random process connecting local maxima of the scalogram along the scale axis and analyzing their properties when the signal fulfills the adaptive harmonic model and is contaminated by stationary Gaussian noise. In addition to establishing several key properties of the AWT for random processes, we investigate the probabilistic characteristics of the resulting random ridge points in the scalogram. Specifically, we establish the uniqueness property of the ridge point at individual time instances and prove the upper hemicontinuity of the ridge random process. Furthermore, we derive bounds on the probability that the deviation between the ridges of noisy and clean signals exceeds a specified threshold, and these bounds depend on the signal-to-noise ratio. To achieve these ridge deviation results, we derive maximal inequalities for the complex modulus of nonstationary Gaussian processes, leveraging classical tools such as the Borell-TIS inequality and Dudley's theorem, which might be of independent interest.

math.ST↗

arXiv2024

Sleep Staging from Airflow Signals Using Fourier Approximations of Persistence Curves

Sleep staging is a challenging task, typically manually performed by sleep technologists based on electroencephalogram and other biosignals of patients taken during overnight sleep studies. Recent work aims to leverage automated algorithms to perform sleep staging not based on electroencephalogram signals, but rather based on the airflow signals of subjects. Prior work uses ideas from topological data analysis (TDA), specifically Hermite function expansions of persistence curves (HEPC) to featurize airflow signals. However, finite order HEPC captures only partial information. In this work, we propose Fourier approximations of persistence curves (FAPC), and use this technique to perform sleep staging based on airflow signals. We analyze performance using an XGBoost model on 1155 pediatric sleep studies taken from the Nationwide Children's Hospital Sleep DataBank (NCHSDB), and find that FAPC methods provide complimentary information to HEPC methods alone, leading to a 4.9% increase in performance over baseline methods.

cs.LG↗

arXiv2024

Landmark Alternating Diffusion

Alternating Diffusion (AD) is a commonly applied diffusion-based sensor fusion algorithm. While it has been successfully applied to various problems, its computational burden remains a limitation. Inspired by the landmark diffusion idea considered in the Robust and Scalable Embedding via Landmark Diffusion (ROSELAND), we propose a variation of AD, called Landmark AD (LAD), which captures the essence of AD while offering superior computational efficiency. We provide a series of theoretical analyses of LAD under the manifold setup and apply it to the automatic sleep stage annotation problem with two electroencephalogram channels to demonstrate its application.

cs.LG↗

arXiv2024

Convergence analysis of t-SNE as a gradient flow for point cloud on a manifold

We present a theoretical foundation regarding the boundedness of the t-SNE algorithm. t-SNE employs gradient descent iteration with Kullback-Leibler (KL) divergence as the objective function, aiming to identify a set of points that closely resemble the original data points in a high-dimensional space, minimizing KL divergence. Investigating t-SNE properties such as perplexity and affinity under a weak convergence assumption on the sampled dataset, we examine the behavior of points generated by t-SNE under continuous gradient flow. Demonstrating that points generated by t-SNE remain bounded, we leverage this insight to establish the existence of a minimizer for KL divergence.

stat.ML↗

arXiv2024

Design a Metric Robust to Complicated High Dimensional Noise for Efficient Manifold Denoising

In this manuscript, we propose an efficient manifold denoiser based on landmark diffusion and optimal shrinkage under the complicated high dimensional noise and compact manifold setup. It is flexible to handle several setups, including the high ambient space dimension with a manifold embedding that occupies a subspace of high or low dimensions, and the noise could be colored and dependent. A systematic comparison with other existing algorithms on both simulated and real datasets is provided. This manuscript is mainly algorithmic and we report several existing tools and numerical results. Theoretical guarantees and more comparisons will be reported in the official paper of this manuscript.

stat.ML↗