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l1-norm regularized l1-norm best-fit lines

Xiao Ling, Paul Brooks

TL;DR

The proposed algorithm demonstrates a worst-case time complexity of $O(n^2 m \log n)$ and, in certain instances, achieves global optimality for the sparse robust subspace, thereby exhibiting polynomial time efficiency.

Abstract

In this work, we propose an optimization framework for estimating a sparse robust one-dimensional subspace. Our objective is to minimize both the representation error and the penalty, in terms of the l1-norm criterion. Given that the problem is NP-hard, we introduce a linear relaxation-based approach. Additionally, we present a novel fitting procedure, utilizing simple ratios and sorting techniques. The proposed algorithm demonstrates a worst-case time complexity of $O(n^2 m \log n)$ and, in certain instances, achieves global optimality for the sparse robust subspace, thereby exhibiting polynomial time efficiency. Compared to extant methodologies, the proposed algorithm finds the subspace with the lowest discordance, offering a smoother trade-off between sparsity and fit. Its architecture affords scalability, evidenced by a 16-fold improvement in computational speeds for matrices of 2000x2000 over CPU version. Furthermore, this method is distinguished by several advantages, including its independence from initialization and deterministic and replicable procedures. Furthermore, this method is distinguished by several advantages, including its independence from initialization and deterministic and replicable procedures. The real-world example demonstrates the effectiveness of algorithm in achieving meaningful sparsity, underscoring its precise and useful application across various domains.

l1-norm regularized l1-norm best-fit lines

TL;DR

The proposed algorithm demonstrates a worst-case time complexity of and, in certain instances, achieves global optimality for the sparse robust subspace, thereby exhibiting polynomial time efficiency.

Abstract

In this work, we propose an optimization framework for estimating a sparse robust one-dimensional subspace. Our objective is to minimize both the representation error and the penalty, in terms of the l1-norm criterion. Given that the problem is NP-hard, we introduce a linear relaxation-based approach. Additionally, we present a novel fitting procedure, utilizing simple ratios and sorting techniques. The proposed algorithm demonstrates a worst-case time complexity of and, in certain instances, achieves global optimality for the sparse robust subspace, thereby exhibiting polynomial time efficiency. Compared to extant methodologies, the proposed algorithm finds the subspace with the lowest discordance, offering a smoother trade-off between sparsity and fit. Its architecture affords scalability, evidenced by a 16-fold improvement in computational speeds for matrices of 2000x2000 over CPU version. Furthermore, this method is distinguished by several advantages, including its independence from initialization and deterministic and replicable procedures. Furthermore, this method is distinguished by several advantages, including its independence from initialization and deterministic and replicable procedures. The real-world example demonstrates the effectiveness of algorithm in achieving meaningful sparsity, underscoring its precise and useful application across various domains.
Paper Structure (12 sections, 5 theorems, 12 equations, 6 figures, 7 tables, 3 algorithms)

This paper contains 12 sections, 5 theorems, 12 equations, 6 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Estimating an $\ell_1$-Norm Regularized $\ell_1$-Norm Best-Fit Line
  3. Experiments with Synthetic Data
  4. A Toy Example
  5. Comparison to Competing Methods
  6. Effect of Varying $m$ and $n$ on Error and Sparsity
  7. Application to Human Microbiome Project Data
  8. Algorithms 1 and 2 on NVIDIA Graphical Processing Units
  9. Introduction
  10. Algorithm \ref{['alg1']} Performance Evaluation
  11. Breakpoints for Algorithm 2
  12. Conclusion

Key Result

Proposition 1

The formulation formulation2 is equivalent to formulation1.

Figures (6)

  • Figure 1: Schematic illustration of breakpoints. Each color depicts the objective function value when preserving a coordinate $\hat{\jmath}$, $z_{\hat{\jmath}}$, as a function of the penalty parameter $\lambda$.
  • Figure 2: The figure displays the discordance (y-axis) versus the $\ell_0$-norm (x-axis) for Algorithm 1 (red), pcaPP (blue), and rosPCA (cyan) on simulated data with increasing levels of contamination. The plots compare the estimated loadings matrix to the true loadings matrix for data with outlier levels ranging from 1% to 6%.
  • Figure 3: Effect on discordance (solid lines) and sparsity (dashed lines) when $\lambda$ is varied for (a) datasets with different $m$ and (b) datasets with different $n$. Sparsity is measured as a percentage of $m$.
  • Figure 4: The number of taxa used for clustering is on the left y-axis for the blue line and average purity is on the right y-axis for the red line.
  • Figure 5: One dimensional decomposition using blocks and threads. The formula will map each thread to an element in the vector.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Proposition 3
  • proof
  • Proposition 4