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Asymptotic Distribution of Bounded Shape Constrained Lasso-Type Estimator for Graph-Structured Signals and Discrete Distributions

Vladimir Pastukhov

TL;DR

The paper studies the asymptotic distribution of a bounded shape constrained lasso-type estimator for graph-structured signals and discrete distributions. It unites fused lasso and nearly-isotonic regression on a directed acyclic graph with sum and range constraints, proving that, under appropriate scaling of penalties, the estimator’s limit is the minimizer of a convex functional $V(\bm{w})$ driven by the base estimator’s limit $\bm{\psi}$; in special cases this reduces to concatenated nearly-isotonic regressions over constant regions. The results show the estimator preserves the convergence rate and exhibits isotonic-regression–like behavior on graphs, with applications to distribution estimation and histogram smoothing on graphs. The framework extends to general DAGs and opens avenues for future work on alternative loss functions and data-driven penalization.

Abstract

This paper is dedicated to the asymptotic distribution of bounded shape constrained lasso-type estimator. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under certain assumptions on the penalization parameters, the limiting distribution of the estimator is given by the certain constrained estimator applied to the asymptotic distribution of the unrestricted estimator, and, consequently, the estimator preserves the rate of convergence of the underlying estimator. Next, without the fusion penalisation term, the limiting distribution of the estimator is given by the concatenation of the individual nearly-isotonic estimators applied to the specific sub-vectors of the asymptotic distribution of the unrestricted estimator. This behaviour is similar to the case of isotonic regression. The obtained results are also applied to the constrained estimation of a discrete distribution supported on a directed acyclic graph.

Asymptotic Distribution of Bounded Shape Constrained Lasso-Type Estimator for Graph-Structured Signals and Discrete Distributions

TL;DR

The paper studies the asymptotic distribution of a bounded shape constrained lasso-type estimator for graph-structured signals and discrete distributions. It unites fused lasso and nearly-isotonic regression on a directed acyclic graph with sum and range constraints, proving that, under appropriate scaling of penalties, the estimator’s limit is the minimizer of a convex functional driven by the base estimator’s limit ; in special cases this reduces to concatenated nearly-isotonic regressions over constant regions. The results show the estimator preserves the convergence rate and exhibits isotonic-regression–like behavior on graphs, with applications to distribution estimation and histogram smoothing on graphs. The framework extends to general DAGs and opens avenues for future work on alternative loss functions and data-driven penalization.

Abstract

This paper is dedicated to the asymptotic distribution of bounded shape constrained lasso-type estimator. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under certain assumptions on the penalization parameters, the limiting distribution of the estimator is given by the certain constrained estimator applied to the asymptotic distribution of the unrestricted estimator, and, consequently, the estimator preserves the rate of convergence of the underlying estimator. Next, without the fusion penalisation term, the limiting distribution of the estimator is given by the concatenation of the individual nearly-isotonic estimators applied to the specific sub-vectors of the asymptotic distribution of the unrestricted estimator. This behaviour is similar to the case of isotonic regression. The obtained results are also applied to the constrained estimation of a discrete distribution supported on a directed acyclic graph.

Paper Structure

This paper contains 6 sections, 3 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Asymptotic distribution of the estimator
  4. Conclusions
  5. Example of partial order relation and corresponding incidence matrix for a graphs
  6. Proofs

Key Result

Theorem 2.1

For any values of the penalization parameters $\lambda^{F}_{n}$ and $\lambda^{NI}_{n}$, the solution to the optimization problem in (FNIGc) is equal to the solution of the following problem i.e., the additional constraints are redundant and automatically satisfied.

Figures (1)

  • Figure 1: Graph $G=(V,E)$ which corresponds to the two dimensional equally spaced grid.

Theorems & Definitions (3)

  • Theorem 2.1
  • Corollary 1
  • Theorem 2.2