Randomized progressive iterative approximation for B-spline curve and surface fittings

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA

Preconditioned geometric iterative methods for cubic B-spline interpolation curves

The geometric iterative method (GIM) is widely used in data interpolation/fitting, but its slow convergence affects the computational efficiency. Recently, much work was done to guarantee the acceleration of GIM in the literature. In this work, we aim to further accelerate the rate of convergence by introducing a preconditioning technique. After constructing the preconditioner, we preprocess the progressive iterative approximation (PIA) and its variants, called the preconditioned GIMs. We show that the proposed preconditioned GIMs converge and the extra computation cost brought by the preconditioning technique is negligible. Several numerical experiments are given to demonstrate that our preconditioner can accelerate the convergence rate of PIA and its variants

Video Data Compression by Progressive Iterative Approximation

In the present paper, the B-spline curve is used for reducing the entropy of video data. We consider the color or luminance variations of a spatial position in a series of frames as input data points in Euclidean space R or R3. The progressive and iterative approximation (PIA) method is a direct and intuitive way of generating curve series of high and higher fitting accuracy. The video data points are approximated using progressive and iterative approximation for least square (LSPIA) fitting. The Lossless video data compression is done through storing the B-spline curve control points (CPs) and the difference between fitted and original video data. The proposed method is applied to two classes of synthetically produced and naturally recorded video sequences and makes a reduction in the entropy of both. However, this reduction is higher for syntactically created than those naturally produced. The comparative analysis of experiments on a variety of video sequences suggests that the entropy of output video data is much less than that of input video data

Progressive Analytics: A Computation Paradigm for Exploratory Data Analysis

Exploring data requires a fast feedback loop from the analyst to the system, with a latency below about 10 seconds because of human cognitive limitations. When data becomes large or analysis becomes complex, sequential computations can no longer be completed in a few seconds and data exploration is severely hampered. This article describes a novel computation paradigm called Progressive Computation for Data Analysis or more concisely Progressive Analytics, that brings at the programming language level a low-latency guarantee by performing computations in a progressive fashion. Moving this progressive computation at the language level relieves the programmer of exploratory data analysis systems from implementing the whole analytics pipeline in a progressive way from scratch, streamlining the implementation of scalable exploratory data analysis systems. This article describes the new paradigm through a prototype implementation called ProgressiVis, and explains the requirements it implies through examples.Comment: 10 page

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