Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of its Fourier transform \hat{1_K}. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k^n) noisy covariogram measurements to construct a convex polytope P_k that approximates K or its reflection -K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between P_k and K or -K tends to zero as k tends to infinity.Comment: Version accepted on the Journal of the American Mathematical Society. With respect to version 1 the noise model has been greatly extended and an appendix has been added, with a discussion of rates of convergence and implementation issues. 56 pages, 4 figure

A Logical Model and Data Placement Strategies for MEMS Storage Devices

MEMS storage devices are new non-volatile secondary storages that have outstanding advantages over magnetic disks. MEMS storage devices, however, are much different from magnetic disks in the structure and access characteristics. They have thousands of heads called probe tips and provide the following two major access facilities: (1) flexibility: freely selecting a set of probe tips for accessing data, (2) parallelism: simultaneously reading and writing data with the set of probe tips selected. Due to these characteristics, it is nontrivial to find data placements that fully utilize the capability of MEMS storage devices. In this paper, we propose a simple logical model called the Region-Sector (RS) model that abstracts major characteristics affecting data retrieval performance, such as flexibility and parallelism, from the physical MEMS storage model. We also suggest heuristic data placement strategies based on the RS model and derive new data placements for relational data and two-dimensional spatial data by using those strategies. Experimental results show that the proposed data placements improve the data retrieval performance by up to 4.0 times for relational data and by up to 4.8 times for two-dimensional spatial data of approximately 320 Mbytes compared with those of existing data placements. Further, these improvements are expected to be more marked as the database size grows.Comment: 37 page

Storage Capacity of Two-dimensional Neural Networks

We investigate the maximum number of embedded patterns in the two-dimensional Hopfield model. The grand state energies of two specific network states, namely, the energies of the pure-ferromagnetic state and the state of specific one stored pattern are calculated exactly in terms of the correlation function of the ferromagnetic Ising model. We also investigate the energy landscape around them by computer simulations. Taking into account the qualitative features of the phase diagrams obtained by Nishimori, Whyte and Sherrington [Phys. Rev. E {\bf 51}, 3628 (1995)], we conclude that the network cannot retrieve more than three patterns.Comment: 13pages, 7figures, revtex