ECE 901 Lecture 15: Denoising Smooth Functions with Unknown Smoothness

ELEN6887 Lecture 15: Denoising Smooth Functions with Unknown Smoothness

Lipschitz functions are interesting, but can be very rough (these can have many kinks). In many situations the functions can be much smoother. This is how you would model the temperature inside a museum room for example. Often we don’t know how smooth the function might be, so an interesting question is if we can adapt to the unknown smoothness. In this lecture we will use the Maximum Complexit...

ELEN6887 Lecture 14: Denoising Smooth Functions with Unknown Smoothness

Lipschitz functions are interesting, but can be very rough (these can have many kinks). In many situations the functions can be much smoother. This is how you would model the temperature inside a museum room for example. Often we don’t know how smooth the function might be, so an interesting question is if we can adapt to the unknown smoothness. In this lecture we will use the Maximum Complexit...

Denoising in Smooth Function Spaces II - Adapting to Unknown Smoothness 3 3 Denoising Example for Signal - plus - Gaussian Noise Observation

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

Consider the following setting. Let Y = f∗(X) +W, where X is a random variable (r.v.) on X = [0, 1], W is a r.v. on Y = R, independent of X and satisfying E[W ] = 0 and E[W ] = σ <∞. Finally let f∗ : [0, 1]→ R be a function satisfying |f∗(t)− f∗(s)| ≤ L|t− s|, ∀t, s ∈ [0, 1], (1) where L > 0 is a constant. A function satisfying condition (1) is said to be Lipschitz on [0, 1]. Notice that such a...

Nonparametric denoising Signals of Unknown Local Structure, II: Nonparametric Regression Estimation

We consider the problem of recovering of continuous multi-dimensional functions f from the noisy observations over the regular grid m−1Zd, m ∈ N∗. Our focus is at the adaptive estimation in the case when the function can be well recovered using a linear filter, which can depend on the unknown function itself. In the companion paper [26] we have shown in the case when there exists an adapted tim...