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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals.

Ishan Wickramasingha1, Ahmed Elrewainy2, Michael Sobhy3

  • 1Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada wickrami@myumanitoba.ca.

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|July 21, 2020
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Tensor Least Angle Regression (T-LARS) offers an efficient solution for large multidimensional sparse least-squares problems. This new method significantly reduces computation time and memory usage compared to Kronecker-OMP, enabling advanced biomedical signal processing.

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Area of Science:

  • Signal Processing
  • Statistical Learning
  • Multidimensional Data Analysis

Background:

  • Sparse signal representation is crucial in signal processing and statistics.
  • Existing methods like OMP, Basis Pursuit, and LARS struggle with large multidimensional problems.
  • Kronecker-OMP addresses multidimensionality but faces scalability issues with memory and computation.

Purpose of the Study:

  • Introduce Tensor Least Angle Regression (T-LARS) as an efficient algorithm for multidimensional sparse least-squares problems.
  • Develop a method that overcomes the computational and memory limitations of existing techniques.
  • Provide a versatile tool for both L1 and L2 constrained sparse recovery in high dimensions.

Main Methods:

  • Generalization of the Least Angle Regression (LARS) algorithm to tensor data.
  • Development of T-LARS for solving large-scale multidimensional sparse least-squares problems.
  • Application of T-LARS to reconstruct 3D brain images using separable dictionaries.

Main Results:

  • T-LARS demonstrates significant speed improvements (46-70x) over Kronecker-OMP for L1-sparse solutions.
  • The algorithm efficiently handles both L1 and L2 constrained problems across all critical regularization parameter values.
  • Numerical experiments successfully applied T-LARS to large 3D brain image sparse representations.

Conclusions:

  • T-LARS provides a computationally efficient and memory-sparing alternative for multidimensional sparse signal processing.
  • While Kronecker-OMP yields slightly lower residual errors for L1-sparsity, T-LARS offers a substantial performance gain.
  • T-LARS is poised to become a valuable tool in various multidimensional biomedical signal processing applications.