Mean Estimation from One-Bit Measurements

We consider the problem of estimating the mean of a symmetric log-concave distribution under the constraint that only a single bit per sample from this distribution is available to the estimator. We study the mean squared error as a function of the sample size (and hence the number of bits). We consider three settings: first, a centralized setting, where an encoder may release $n$ bits given a sample of size $n$, and for which there is no asymptotic penalty for quantization; second, an adaptive setting in which each bit is a function of the current observation and previously recorded bits, where we show that the optimal relative efficiency compared to the sample mean is precisely the efficiency of the median; lastly, we show that in a distributed setting where each bit is only a function of a local sample, no estimator can achieve optimal efficiency uniformly over the parameter space. We additionally complement our results in the adaptive setting by showing that \emph{one} round of adaptivity is sufficient to achieve optimal mean-square error

Mean Estimation from Adaptive One-bit Measurements

We consider the problem of estimating the mean of a normal distribution under the following constraint: the estimator can access only a single bit from each sample from this distribution. We study the squared error risk in this estimation as a function of the number of samples and one-bit measurements $n$. We consider an adaptive estimation setting where the single-bit sent at step $n$ is a function of both the new sample and the previous $n-1$ acquired bits. For this setting, we show that no estimator can attain asymptotic mean squared error smaller than $\pi/(2n)+O(n^{-2})$ times the variance. In other words, one-bit restriction increases the number of samples required for a prescribed accuracy of estimation by a factor of at least $\pi/2$ compared to the unrestricted case. In addition, we provide an explicit estimator that attains this asymptotic error, showing that, rather surprisingly, only $\pi/2$ times more samples are required in order to attain estimation performance equivalent to the unrestricted case

Distortion-Rate Function of Sub-Nyquist Sampled Gaussian Sources

The amount of information lost in sub-Nyquist sampling of a continuous-time Gaussian stationary process is quantified. We consider a combined source coding and sub-Nyquist reconstruction problem in which the input to the encoder is a noisy sub-Nyquist sampled version of the analog source. We first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate-limited version of its samples. This expression is a function of the sampling frequency and the average number of bits describing each sample. It is given as the sum of two terms: Minimum mean square error in estimating the source from its noisy but otherwise fully observed sub-Nyquist samples, and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source. We extend this result to multi-branch uniform sampling, where the samples are available through a set of parallel channels with a uniform sampler and a pre-sampling filter in each branch. Further optimization to reduce distortion is then performed over the pre-sampling filters, and an optimal set of pre-sampling filters associated with the statistics of the input signal and the sampling frequency is found. This results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering. These results thus unify the Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion theory for Gaussian sources.Comment: Accepted for publication at the IEEE transactions on information theor

A note on a local ergodic theorem for an infinite tower of coverings

This is a note on a local ergodic theorem for a symmetric exclusion process defined on an infinite tower of coverings, which is associated with a finitely generated residually finite amenable group.Comment: Final version to appear in Springer Proceedings in Mathematics and Statistic

Maximum likelihood reconstruction for Ising models with asynchronous updates

We describe how the couplings in an asynchronous kinetic Ising model can be inferred. We consider two cases, one in which we know both the spin history and the update times and one in which we only know the spin history. For the first case, we show that one can average over all possible choices of update times to obtain a learning rule that depends only on spin correlations and can also be derived from the equations of motion for the correlations. For the second case, the same rule can be derived within a further decoupling approximation. We study all methods numerically for fully asymmetric Sherrington-Kirkpatrick models, varying the data length, system size, temperature, and external field. Good convergence is observed in accordance with the theoretical expectations

Entropy and efficiency of a molecular motor model

In this paper we investigate the use of path-integral formalism and the concepts of entropy and traffic in the context of molecular motors. We show that together with time-reversal symmetry breaking arguments one can find bounds on efficiencies of such motors. To clarify this techinque we use it on one specific model to find both the thermodynamic and the Stokes efficiencies, although the arguments themselves are more general and can be used on a wide class of models. We also show that by considering the molecular motor as a ratchet, one can find additional bounds on the thermodynamic efficiency

Hydrodynamic limit for a boundary driven stochastic lattice gas model with many conserved quantities

We prove the hydrodynamic limit for a particle system in which particles may have different velocities. We assume that we have two infinite reservoirs of particles at the boundary: this is the so-called boundary driven process. The dynamics we considered consists of a weakly asymmetric simple exclusion process with collision among particles having different velocities