# Weighted reverse Poincare inequality over a function class of neural networks

We consider a probability measure supported on the whole space $$\mathbb{R}^n$$, whose density is $$p(x)$$. We also consider a (one-layer) neural network function class $$\mathcal{C}$$, whose elements have the form $$f(x)=\sum_{i=1}^rv_i\sigma(w_i^{\top}x).\tag{neural network}$$ Here $$r$$ is the number of nodes in the hidden layer, $$\sigma$$ is the active function (such as ReLu, sigmoid), $$v_i$$ and $$w_i$$ are parameters. Such a class can also be viewed as a discretization of the integral $$g(x) = \int_t v(t) \sigma(w(t)^\top x) d t,\tag{continuous version}$$ which is common in harmonic analysis. A familiar example is when $$\sigma$$ is a trigonometric function and $$v(t)$$ is a constant, it resembles the Fourier integral, where $$w(t)$$ is the frequency.

My question is that, under what conditions on the probability measure $$p(x)$$ and the structure of function class $$\sigma, r,v_i,w_i$$, weighted reverse Poincare inequality holds over $$\mathcal{C}$$? That is to say, there exists a constant $$K$$, such that for any $$f,g \in \mathcal(C)$$, the following inequality holds. $$\int_{\mathbb{R}^n}\|\nabla f(x)-\nabla g(x)\|^2 p(x) dx\le K\int_{\mathbb{R}^n}| f(x)-g(x)|^2 p(x)dx.%\tag{reverse Poincare}$$
In other words, we are able to upper bound the difference in gradients with the difference in function values (which is in the reverse direction of the commonly known Poincare inequality).