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Fused $\ell_{1}$ Trend Filtering on Graphs

Vladimir Pastukhov

TL;DR

This paper shows how fused estimator and 1-st order trend filtering are related to each other and proposes a computationally feasible numerical solution with a linear complexity per iteration with respect to the amount of edges in the graph.

Abstract

This paper is dedicated to the fused trend filtering on a general graph, which is a combination of fused estimator and 1-st order trend filtering on a graph. There are two cases of fusion regularisers studied in this work: anisotropic total variation (i.e. fused lasso) and nearly-isotonic restriction. For the trend filtering part we consider general trend filtering on a given graph and Kronecker trend filter for the case of lattice data. We show how these estimators are related to each other and propose a computationally feasible numerical solution with a linear complexity per iteration with respect to the amount of edges in the graph.

Fused $\ell_{1}$ Trend Filtering on Graphs

TL;DR

This paper shows how fused estimator and 1-st order trend filtering are related to each other and proposes a computationally feasible numerical solution with a linear complexity per iteration with respect to the amount of edges in the graph.

Abstract

This paper is dedicated to the fused trend filtering on a general graph, which is a combination of fused estimator and 1-st order trend filtering on a graph. There are two cases of fusion regularisers studied in this work: anisotropic total variation (i.e. fused lasso) and nearly-isotonic restriction. For the trend filtering part we consider general trend filtering on a given graph and Kronecker trend filter for the case of lattice data. We show how these estimators are related to each other and propose a computationally feasible numerical solution with a linear complexity per iteration with respect to the amount of edges in the graph.
Paper Structure (8 sections, 4 theorems, 70 equations, 6 figures)

This paper contains 8 sections, 4 theorems, 70 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. General statement of the problem
  4. Organisation of the paper
  5. Solution to the fused $\ell_{1}$ trend filtering
  6. Computational aspects, simulation study and application to the real data
  7. Conclusion and discussion
  8. Examples of partial order relation and matrices for the graphs

Key Result

Theorem 2.1

For a fixed data vector $\bm{y} \in \mathbb{R}^{n}$ indexed by the index set $\mathcal{I}$ with the partial order relation $\preceq$ defined on $\mathcal{I}$ and the penalisation parameters $\lambda_{NI}$, $\lambda_{F}$ and $\lambda_{T}$ the solution to the nearly-isotonic trend filtering in (NITFG) with subject to and solution to the fused lasso trend filtering is with subject to where $D$

Figures (6)

  • Figure 1: Computational times vs side size of a square grid for ADMM and OSQP algorithms for fused lasso trend filtering for the cases of general trend filtering and Kronecker trend filtering in two dimensional grid.
  • Figure 2: Trend filter and Kronecker trend filter for different values of parameter $\lambda_{T}$.
  • Figure 3: Isotonic regression and nearly-isotonic Kronecker trend filter for different values of parameter $\lambda_{NI}$ and $\lambda_{T}$.
  • Figure 4: Original image, image with missing pixels and noise, and different image filters.
  • Figure 5: Graph $G=(V,E)$ which corresponds to the chain graph.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • proof