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We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented…
Multi-index models provide a popular framework to investigate the learnability of functions with low-dimensional structure and, also due to their connections with neural networks, they have been object of recent intensive study. In this…
We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say $x_0\in \mathbb{C}^d$) are…
Inverse scattering methods capable of compressive imaging are proposed and analyzed. The methods employ randomly and repeatedly (multiple-shot) the single-input-single-output (SISO) measurements in which the probe frequencies, the incident…
Dimension reduction algorithms are a crucial part of many data science pipelines, including data exploration, feature creation and selection, and denoising. Despite their wide utilization, many non-linear dimension reduction algorithms are…
This paper explores and analyzes two randomized designs for robust Principal Component Analysis (PCA) employing low-dimensional data sketching. In one design, a data sketch is constructed using random column sampling followed by low…
Distributed algorithms and theories are called for in this era of big data. Under weaker local signal-to-noise ratios, we improve upon the celebrated one-round distributed principal component analysis (PCA) algorithm designed in the spirit…
Principal component analysis (PCA) is perhaps the most widely used method for data dimensionality reduction. A key question in PCA is deciding how many factors to retain. This manuscript describes a new approach to automatically selecting…
The high-dimensional data setting, in which p >> n, is a challenging statistical paradigm that appears in many real-world problems. In this setting, learning a compact, low-dimensional representation of the data can substantially help…
Distributed computing is a standard way to scale up machine learning and data science algorithms to process large amounts of data. In such settings, avoiding communication amongst machines is paramount for achieving high performance. Rather…
Line spectral estimation theory aims to estimate the off-the-grid spectral components of a time signal with optimal precision. Recent results have shown that it is possible to recover signals having sparse line spectra from few temporal…
This paper introduces a popular dimension reduction method, sliced inverse regression (SIR), into multivariate statistical process monitoring. Provides an extension of SIR for the single-index model by adopting the idea from partial least…
This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is…
The dimensional reductions in the branched polymer and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. The small size minors are applied for the evaluations of the scale dimensions of these two models and…
We propose a class of dimension reduction methods for right censored survival data using a counting process representation of the failure process. Semiparametric estimating equations are constructed to estimate the dimension reduction…
Stationary subspace analysis (SSA) is a blind source separation framework that decomposes linearly mixed multivariate data into stationary and nonstationary components. We extend SSA to spatially indexed data by introducing spatial…
The Principal Component Analysis (PCA) is a data dimensionality reduction technique well-suited for processing data from sensor networks. It can be applied to tasks like compression, event detection, and event recognition. This technique is…
In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables and parameters, the Buckingham Pi theorem provides a procedure…
Two major bottlenecks to the solution of large-scale Bayesian inverse problems are the scaling of posterior sampling algorithms to high-dimensional parameter spaces and the computational cost of forward model evaluations. Yet incomplete or…
The real-life data have a complex and non-linear structure due to their nature. These non-linearities and the large number of features can usually cause problems such as the empty-space phenomenon and the well-known curse of dimensionality.…