DROP: Dimensionality Reduction Optimization for Time Series

Dimensionality reduction is a critical step in scaling machine learning pipelines. Principal component analysis (PCA) is a standard tool for dimensionality reduction, but performing PCA over a full dataset can be prohibitively expensive. As a result, theoretical work has studied the effectiveness of iterative, stochastic PCA methods that operate over data samples. However, termination conditions for stochastic PCA either execute for a predetermined number of iterations, or until convergence of the solution, frequently sampling too many or too few datapoints for end-to-end runtime improvements. We show how accounting for downstream analytics operations during DR via PCA allows stochastic methods to efficiently terminate after operating over small (e.g., 1%) subsamples of input data, reducing whole workload runtime. Leveraging this, we propose DROP, a DR optimizer that enables speedups of up to 5x over Singular-Value-Decomposition-based PCA techniques, and exceeds conventional approaches like FFT and PAA by up to 16x in end-to-end workloads

Dynamic Algorithms and Asymptotic Theory for Lp-norm Data Analysis

The focus of this dissertation is the development of outlier-resistant stochastic algorithms for Principal Component Analysis (PCA) and the derivation of novel asymptotic theory for Lp-norm Principal Component Analysis (Lp-PCA). Modern machine learning and signal processing applications employ sensors that collect large volumes of data measurements that are stored in the form of data matrices, that are often massive and need to be efficiently processed in order to enable machine learning algorithms to perform effective underlying pattern discovery. One such commonly used matrix analysis technique is PCA. Over the past century, PCA has been extensively used in areas such as machine learning, deep learning, pattern recognition, and computer vision, just to name a few. PCA\u27s popularity can be attributed to its intuitive formulation on the L2-norm, availability of an elegant solution via the singular-value-decomposition (SVD), and asymptotic convergence guarantees. However, PCA has been shown to be highly sensitive to faulty measurements (outliers) because of its reliance on the outlier-sensitive L2-norm. Arguably, the most straightforward approach to impart robustness against outliers is to replace the outlier-sensitive L2-norm by the outlier-resistant L1-norm, thus formulating what is known as L1-PCA. Exact and approximate solvers are proposed for L1-PCA in the literature. On the other hand, in this big-data era, the data matrix may be very large and/or the data measurements may arrive in streaming fashion. Traditional L1-PCA algorithms are not suitable in this setting. In order to efficiently process streaming data, while being resistant against outliers, we propose a stochastic L1-PCA algorithm that computes the dominant principal component (PC) with formal convergence guarantees. We further generalize our stochastic L1-PCA algorithm to find multiple components by propose a new PCA framework that maximizes the recently proposed Barron loss. Leveraging Barron loss yields a stochastic algorithm with a tunable robustness parameter that allows the user to control the amount of outlier-resistance required in a given application. We demonstrate the efficacy and robustness of our stochastic algorithms on synthetic and real-world datasets. Our experimental studies include online subspace estimation, classification, video surveillance, and image conditioning, among other things. Last, we focus on the development of asymptotic theory for Lp-PCA. In general, Lp-PCA for p\u3c2 has shown to outperform PCA in the presence of outliers owing to its outlier resistance. However, unlike PCA, Lp-PCA is perceived as a ``robust heuristic\u27\u27 by the research community due to the lack of theoretical asymptotic convergence guarantees. In this work, we strive to shed light on the topic by developing asymptotic theory for Lp-PCA. Specifically, we show that, for a broad class of data distributions, the Lp-PCs span the same subspace as the standard PCs asymptotically and moreover, we prove that the Lp-PCs are specific rotated versions of the PCs. Finally, we demonstrate the asymptotic equivalence of PCA and Lp-PCA with a wide variety of experimental studies

A new SVD approach to optimal topic estimation

In the probabilistic topic models, the quantity of interest---a low-rank matrix consisting of topic vectors---is hidden in the text corpus matrix, masked by noise, and Singular Value Decomposition (SVD) is a potentially useful tool for learning such a matrix. However, different rows and columns of the matrix are usually in very different scales and the connection between this matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges. We overcome the challenges by introducing a proper Pre-SVD normalization of the text corpus matrix and a proper column-wise scaling for the matrix of interest, and by revealing a surprising Post-SVD low-dimensional {\it simplex} structure. The simplex structure, together with the Pre-SVD normalization and column-wise scaling, allows us to conveniently reconstruct the matrix of interest, and motivates a new SVD-based approach to learning topic models. We show that under the popular probabilistic topic model \citep{hofmann1999}, our method has a faster rate of convergence than existing methods in a wide variety of cases. In particular, for cases where documents are long or $n$ is much larger than $p$, our method achieves the optimal rate. At the heart of the proofs is a tight element-wise bound on singular vectors of a multinomially distributed data matrix, which do not exist in literature and we have to derive by ourself. We have applied our method to two data sets, Associated Process (AP) and Statistics Literature Abstract (SLA), with encouraging results. In particular, there is a clear simplex structure associated with the SVD of the data matrices, which largely validates our discovery.Comment: 73 pages, 8 figures, 6 tables; considered two different VH algorithm, OVH and GVH, and provided theoretical analysis for each algorithm; re-organized upper bound theory part; added the subsection of comparing error rate with other existing methods; provided another improved version of error analysis through Bernstein inequality for martingale