Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On regularized polynomial functional regression

Markus Holzleitner, Sergei Pereverzyev

TL;DR

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound, which extends and generalizes several findings from the context of linear functional regression as well.

Abstract

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.

On regularized polynomial functional regression

TL;DR

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound, which extends and generalizes several findings from the context of linear functional regression as well.

Abstract

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.
Paper Structure (8 sections, 14 theorems, 100 equations, 4 figures)

This paper contains 8 sections, 14 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Overall setting and assumptions
  4. Operator norms and related auxiliary estimates
  5. Statement and proof of main result
  6. Computations for Tikhonov regularization and its iterations
  7. Experimental evaluation
  8. Acknowledgements

Key Result

Lemma 1

Let $\text{HS}(\mathcal{H}_1, \mathcal{H}_2)$ denote the Hilbert space of Hilbert-Schmidt operators between Hilbert spaces $\mathcal{H}_1$ and $\mathcal{H}_2$. For simplicity let us also use $\text{HS}(\mathcal{H}_1, \mathcal{H}_1)=\text{HS}(\mathcal{H}_1).$ Under Assumption ass:unif for $1 \le k,l

Figures (4)

  • Figure 1: Error curve for $\lambda=10^{-1}$
  • Figure 2: Error curve for $\lambda=10^{-3}$
  • Figure 3: Error curve for $\lambda=10^{-9}$
  • Figure 4: Error curve for $\lambda=10^{-9}$ with additive Gaussian noise.

Theorems & Definitions (29)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6: pinelis1994optimum
  • Lemma 7: see e.g. Theorem 3.3.4. in yurinsky1995sums
  • proof : Proof of Lemma \ref{['lem:op_est_0']}
  • proof : Proof of Lemma \ref{['lem:op_est_1']}
  • ...and 19 more