Infinite-range transverse field Ising models and quantum computation

We present a brief review on information processing, computing and inference via quantum fluctuation, and clarify the relationship between the probabilistic information processing and theory of quantum spin glasses through the analysis of the infinite-range model. We also argue several issues to be solved for the future direction in the research field.Comment: 13 pages, 6 figures, using svjour.cls, to appear in EPJ-Special Topic

Modelling and estimation for random fields

Caption title.Includes bibliographical references (p. [21]-[22]).Supported by Air Force Office of Scientific Research. AFOSR-89-0276-C Supported by the Army Research Office. DAAL03-92-G-0115Sanjoy K. Mitter

Consistencies and rates of convergence of jump-penalized least squares estimators

We study the asymptotics for jump-penalized least squares regression aiming at approximating a regression function by piecewise constant functions. Besides conventional consistency and convergence rates of the estimates in $L^2([0,1))$ our results cover other metrics like Skorokhod metric on the space of c\`{a}dl\`{a}g functions and uniform metrics on $C([0,1])$. We will show that these estimators are in an adaptive sense rate optimal over certain classes of "approximation spaces." Special cases are the class of functions of bounded variation (piecewise) H\"{o}lder continuous functions of order $0<\alpha\le1$ and the class of step functions with a finite but arbitrary number of jumps. In the latter setting, we will also deduce the rates known from change-point analysis for detecting the jumps. Finally, the issue of fully automatic selection of the smoothing parameter is addressed.Comment: Published in at http://dx.doi.org/10.1214/07-AOS558 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On sampling methods and annealing algorithms

Includes bibliographical references (p. 12-14).Cover title.Research supported by the National Science Foundation. ECS-8910073 Research supported by the Air Force Office of Scientific Research. 89-0276B Research supported by the Army Research Office. DAAL03-86-K-0171Saul B. Gelfand and Sanjoy K. Mitter