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Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $$K$$ and $$L$$ be star bodies in $${\mathbb R}^n,$$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $$K$$ and $$L$$, respectively, so that $$\|g\|_\infty =g(0)=1.$$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $$C$$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of $${\mathbb R}^n,$$ and $$d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$$ is the outer volume ratio distance from $$K$$ to the class of generalized $$k$$-intersection bodies in $${\mathbb R}^n.$$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem.
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Mean Field Analysis of Sparse Reconstruction with Correlated Variables.
Sparse reconstruction algorithms aim to retrieve high-dimensional sparse signals from a limited number of measurements. A common example is LASSO or Basis Pursuit where sparsity is enforced using an `1-penalty together with a cost function ||y − Hx||_2^2. For random design matrices H, a sharp phase transition boundary separates the ‘good’ parameter region where error-free recovery of a sufficiently sparse signal is possible and a ‘bad’ regime where the recovery fails. However, theoretical analysis of phase transition boundary of the correlated variables case lags behind that of uncorrelated variables. Here we use replica trick from statistical physics to show that when an Ndimensional signal x is K-sparse and H is M × N dimensional with the covariance E[H_{ia}H_{jb}] = 1 M C_{ij}D_{ab}, with all D_{aa} = 1, the perfect recovery occurs at M ∼ ψ_K(D)K log(N/M) in the very sparse limit, where ψ_K(D) ≥ 1, indicating need for more observations for the same degree of sparsity.
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- Award ID(s):
- 1344069
- PAR ID:
- 10019671
- Date Published:
- Journal Name:
- Eusipco
- ISSN:
- 2076-1465
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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