The design of sampling set (DoS) for bandlimited graph signals (GS) has been extensively studied in recent years, but few of them exploit the benefits of the stochastic prior of GS. In this work, we introduce the optimization framework for Bayesian DoS of bandlimited GS. We also illustrate how the choice of different sampling sets affects the estimation error and how the prior knowledge influences the result of DoS compared with the non-Bayesian DoS by the aid of analyzing Gershgorin discs of error metric matrix. Finally, based on our analysis, we propose a heuristic algorithm for DoS to avoid solving the optimization problem directly.
In this paper, we mainly investigate the nonuniform sampling for random signals which are bandlimited in the linear canonical transform (LCT) domain. We show that the nonuniform sampling for a random signal bandlimited in the LCT domain is equal to the uniform sampling in the sense of second order statistic characters after a pre-filter in the LCT domain. Moreover, we propose an approximate recovery approach for nonuniform sampling of random signals bandlimited in the LCT domain. Furthermore, we study the mean square error of the nonuniform sampling. Finally, we do some simulations to verify t…
This paper investigates the active sampling for estimation of approximately bandlimited graph signals. With the assistance of a graph filter, an approximately bandlimited graph signal can be formulated by a Gaussian random field over the graph. In contrast to offline sampling set design methods which usually rely on accurate prior knowledge about the model, unknown parameters in signal and noise distribution are allowed in the proposed active sampling algorithm. The active sampling process is divided into two alternating stages: unknown parameters are first estimated by Expectation Maximizatio…
Optimality results for two outstanding Bayesian estimation problems are given in this paper: the estimation of the sampling distribution for the squared total variation function and the estimation of the density for the $L^1$-squared loss function. The posterior predictive distribution provides the solution to these problems. Some examples are presented to illustrate it. The Bayesian estimation problem of a distribution function is also addressed. Consistency of the estimator of the density is proved.
A continuous-time graph signal can be viewed as a time series of graph signals. It generalizes both the classical continuous-time signal and ordinary graph signal. Therefore, such a signal can be considered as a function on two domains: the graph domain and the time domain. In this paper, we consider the sampling theory of bandlimited continuous-time graph signals. To formulate the sampling problem, we need to consider the interaction between the graph and time domains. We describe an explicit procedure to determine a discrete sampling set for perfect signal recovery. Moreover, in analogous to…
We study the problem of sampling and reconstruction of bandlimited graph signals where the objective is to select a node subset of prescribed cardinality that ensures interpolation of the original signal with the lowest reconstruction error. We propose an efficient iterative selection sampling approach and show that in the noiseless case the original signal is exactly recovered from the set of selected nodes. In the case of noisy measurements, a bound on the reconstruction error of the proposed algorithm is established. We further address the support identification of the bandlimited signal wi…
We derive explicit Bayesian nonparametric analysis for a species sampling model with finitely many types of Gibbs form of type $α= -1$ recently introduced in Gnedin (2009). Our results complement existing analysis under Gibbs priors of type $α\in [0, 1)$ proposed in Lijoi et al. (2008). Calculations rely on a groups sequential construction of Gibbs partitions introduced in Cerquetti (2008).
When sampling multiple signals, the correlation between the signals can be exploited to reduce the overall number of samples. In this paper, we study the sampling theory of multiple correlated signals, using correlation to sample them at the lowest sampling rate. Based on the correlation between signal sources, we model multiple continuous-time signals as continuous time-vertex graph signals. The graph signals are projected onto orthogonal bases to remove spatial correlation and reduce dimensions by graph Fourier transform. When the bandwidths of the original signals and the reduced dimension …
We investigate time encoding as an alternative method to classical sampling, and address the problem of reconstructing classes of non-bandlimited signals from time-based samples. We consider a sampling mechanism based on first filtering the input, before obtaining the timing information using a time encoding machine. Within this framework, we show that sampling by timing is equivalent to a non-uniform sampling problem, where the reconstruction of the input depends on the characteristics of the filter and on its non-uniform shifts. The classes of filters we focus on are exponential and polynomi…
It is of particular interest to reconstruct or estimate bandlimited graph signals, which are smoothly varying signals defined over graphs, from partial noisy measurements. However, choosing an optimal subset of nodes to sample is NP-hard. We formularize the problem as the experimental design of a linear regression model if we allow multiple measurements on a single node. By relaxing it to a convex optimization problem, we get the proportion of sample for each node given the budget of total sample size. Then, we use a probabilistic quantization to get the number of each node to be sampled. More…
Many engineering, social, and biological complex systems consist of dynamical elements connected via a large-scale network. Monitoring the network's dynamics is essential for a variety of maintenance and scientific purposes. Whilst we understand how to optimally sample a single dynamic element or a non-dynamic graph, we do not possess a theory on how to optimally sample networked dynamical elements. Here, we study nonlinear dynamic graph signals on a fixed complex network. We define the necessary conditions for optimal sampling in the combining time- and graph-domain to fully recover the netwo…
Conventional sampling focuses on encoding and decoding bandlimited signals by recording signal amplitudes at known time points. Alternately, sampling can be approached using biologically-inspired schemes. Among these are integrate-and-fire time encoding machines (IF-TEMs). They behave like simplified versions of spiking neurons and encode their input using spike times rather than amplitudes. Moreover, when multiple of these neurons jointly process a set of mixed signals, they form one layer in a feedforward spiking neural network. In this paper, we investigate the encoding and decoding potenti…
Sampling is classically performed by recording the amplitude of an input signal at given time instants; however, sampling and reconstructing a signal using multiple devices in parallel becomes a more difficult problem to solve when the devices have an unknown shift in their clocks. Alternatively, one can record the times at which a signal (or its integral) crosses given thresholds. This can model integrate-and-fire neurons, for example, and has been studied by Lazar and Tóth under the name of ``Time Encoding Machines''. This sampling method is closer to what is found in nature. In this paper, …
The recovery of bandlimited signals with high dynamic range is a hot issue in sampling research. The unlimited sampling theory expands the recordable range of traditional analog-to-digital converters (ADCs) arbitrarily, and the signal is folded back into a low dynamic range measurement, avoiding the saturation problem. We study the unlimited sampling problem of high dynamic non-bandlimited signals in the Fourier domain (FD) based on the fractional Fourier transform (FRFT). First, a mathematical signal model for unlimited sampling is proposed. Then, based on this mathematical model, the annihil…
We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=ϕ_t\ast f$ resulting from the convolution with a kernel $ϕ_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We prov…
The R package BiDAG implements Markov chain Monte Carlo (MCMC) methods for structure learning and sampling of Bayesian networks. The package includes tools to search for a maximum a posteriori (MAP) graph and to sample graphs from the posterior distribution given the data. A new hybrid approach to structure learning enables inference in large graphs. In the first step, we define a reduced search space by means of the PC algorithm or based on prior knowledge. In the second step, an iterative order MCMC scheme proceeds to optimize within the restricted search space and estimate the MAP graph. Sa…
Recent advances in deep reinforcement learning have made significant strides in performance on applications such as Go and Atari games. However, developing practical methods to balance exploration and exploitation in complex domains remains largely unsolved. Thompson Sampling and its extension to reinforcement learning provide an elegant approach to exploration that only requires access to posterior samples of the model. At the same time, advances in approximate Bayesian methods have made posterior approximation for flexible neural network models practical. Thus, it is attractive to consider a…
We study the problem of sampling and reconstructing spectrally sparse graph signals where the objective is to select a subset of nodes of prespecified cardinality that ensures interpolation of the original signal with the lowest possible reconstruction error. This task is of critical importance in Graph signal processing (GSP) and while existing methods generally provide satisfactory performance, they typically entail a prohibitive computational cost when it comes to the study of large-scale problems. Thus, there is a need for accelerated and efficient methods tailored for high-dimensional and…
Target encoding is an effective encoding technique of categorical variables and is often used in machine learning systems for processing tabular data sets with mixed numeric and categorical variables. Recently en enhanced version of this encoding technique was proposed by using conjugate Bayesian modeling. This paper presents a further development of Bayesian encoding method by using sampling techniques, which helps in extracting information from intra-category distribution of the target variable, improves generalization and reduces target leakage.
One of the most crucial challenges in graph signal processing is the sampling of bandlimited graph signals, i.e., signals that are sparse in a well-defined graph Fourier domain. So far, the prior art is mostly focused on (sub)sampling single snapshots of graph signals ignoring their evolution over time. However, time can bring forth new insights, since many real signals like sensor measurements, biological, financial, and network signals in general, have intrinsic correlations in both domains. In this work, {we fill this lacuna} by jointly considering the graph-time nature of graph signals, na…
Cluster sampling is common in survey practice, and the corresponding inference has been predominantly design-based. We develop a Bayesian framework for cluster sampling and account for the design effect in the outcome modeling. We consider a two-stage cluster sampling design where the clusters are first selected with probability proportional to cluster size, and then units are randomly sampled inside selected clusters. Challenges arise when the sizes of nonsampled cluster are unknown. We propose nonparametric and parametric Bayesian approaches for predicting the unknown cluster sizes, with thi…
University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
The problem of adaptive sampling for estimating probability mass functions (pmf) uniformly well is considered. Performance of the sampling strategy is measured in terms of the worst-case mean squared error. A Bayesian variant of the existing upper confidence bound (UCB) based approaches is proposed. It is shown analytically that the performance of this Bayesian variant is no worse than the existing approaches. The posterior distribution on the pmfs in the Bayesian setting allows for a tighter computation of upper confidence bounds which leads to significant performance gains in practice. Using…
In the case of informative sampling the sampling scheme explicitly or implicitly depends on the response variable. As a result, the sample distribution of response variable can- not be used for making inference about the population. In this research I investigate the problem of informative sampling from the Bayesian perspective. Application of the Bayesian approach permits solving the problems, which arise due to complexity of the models, being used for handling informative sampling. The main objective of the re- search is to combine the elements of the classical sampling theory and Bayesian a…
Radial basis functions are a common mathematical tool used to construct a smooth interpolating function from a set of data points. A spatial prior based on thin-plate spline radial basis functions can be easily implemented resulting in a posterior that can be sampled directly using Monte Carlo integration, avoiding the computational burden and potential inefficiency of an Monte Carlo Markov Chain (MCMC) sampling scheme. The derivation of the prior and sampling scheme are demonstrated.
Conventional sampling and interpolation commonly rely on discrete measurements. In this paper, we develop a theoretical framework for extrapolation of signals in higher dimensions from knowledge of the continuous waveform on bounded high-dimensional regions. In particular, we propose an iterative method to reconstruct bandlimited multidimensional signals based on truncated versions of the original signal to bounded regions---herein referred to as continuous measurements. In the proposed method, the reconstruction is performed by iterating on a convex combination of region-limiting and bandlimi…
In this paper, we consider multi-channel sampling (MCS) for graph signals. We generally encounter full-band graph signals beyond the bandlimited one in many applications, such as piecewise constant/smooth and union of bandlimited graph signals. Full-band graph signals can be represented by a mixture of multiple signals conforming to different generation models. This requires the analysis of graph signals via multiple sampling systems, i.e., MCS, while existing approaches only consider single-channel sampling. We develop a MCS framework based on generalized sampling. We also present a sampling …
Achieving high resolution time-of-arrival (TOA) estimation in multipath propagation scenarios from bandlimited observations of communication signals is challenging because the multipath channel impulse response (CIR) is not bandlimited. Modeling the CIR as a sparse sequence of Diracs, TOA estimation becomes a problem of parametric spectral inference from observed bandlimited signals. To increase resolution without arriving at unrealistic sampling rates, we consider multiband sampling approach, and propose a practical multibranch receiver for the acquisition. The resulting data model exhibits m…
Graph signal processing (GSP) studies signals that live on irregular data kernels described by graphs. One fundamental problem in GSP is sampling---from which subset of graph nodes to collect samples in order to reconstruct a bandlimited graph signal in high fidelity. In this paper, we seek a sampling strategy that minimizes the mean square error (MSE) of the reconstructed bandlimited graph signals assuming an independent and identically distributed (iid) noise model---leading naturally to the A-optimal design criterion. To avoid matrix inversion, we first prove that the inverse of the informa…
The Unlimited Sensing Framework (USF) is a digital acquisition protocol that allows for sampling and reconstruction of high dynamic range signals. By acquiring modulo samples, the USF circumvents the clipping or saturation problem that is a fundamental bottleneck in conventional analog-to-digital converters (ADCs). In the context of the USF, several works have focused on bandlimited function classes and recently, a hardware validation of the modulo sampling approach has been presented. In a different direction, in this paper we focus on non-bandlimited function classes and consider the well-kn…
Analog to digital converters (ADCs) act as a bridge between the analog and digital domains. Two important attributes of any ADC are sampling rate and its dynamic range. For bandlimited signals, the sampling should be above the Nyquist rate. It is also desired that the signals' dynamic range should be within that of the ADC's; otherwise, the signal will be clipped. Nonlinear operators such as modulo or companding can be used prior to sampling to avoid clipping. To recover the true signal from the samples of the nonlinear operator, either high sampling rates are required or strict constraints on…
We consider the problem of recovering a continuous-time bandlimited signal from the discrete-time signal obtained from sampling it every $T_s$ seconds and reducing the result modulo $Δ$, for some $Δ>0$. For $Δ=\infty$ the celebrated Shannon-Nyquist sampling theorem guarantees that perfect recovery is possible provided that the sampling rate $1/T_s$ exceeds the so-called Nyquist rate. Recent work by Bhandari et al. has shown that for any $Δ>0$ perfect reconstruction is still possible if the sampling rate exceeds the Nyquist rate by a factor of $πe$. In this letter we improve upon this result an…
A signal space approach is presented to study the Nyquist sampling, number of degrees of freedom and reconstruction of an electromagnetic field under arbitrary scattering conditions. Conventional signal processing tools, such as the multidimensional sampling theorem and Fourier theory, are used to provide a linear system theoretic interpretation of electromagnetic wave propagation, thereby revealing the spatially bandlimited nature of electromagnetic fields. Their spatial bandwidth is dictated by the selectivity of the underlying scattering that allows establishing the Nyquist spatial sampling…
This paper investigates the problem of sampling and reconstructing bandpass signals using time encoding machine(TEM). It is shown that the sampling in principle is equivalent to periodic non-uniform sampling (PNS). Then the TEM parameters can be set according to the signal bandwidth and amplitude instead of upper-edge frequency and amplitude as in the case of bandlimited/lowpass signals. For a bandpass signal of a single information band, it can be perfectly reconstructed if the TEM parameters are such that the difference between any consecutive values of the time sequence in each channel is b…
The aim of this chapter is to give an overview of the recent advances related to sampling and recovery of signals defined over graphs. First, we illustrate the conditions for perfect recovery of bandlimited graph signals from samples collected over a selected set of vertexes. Then, we describe some sampling design criteria proposed in the literature to mitigate the effect of noise and model mismatching when performing graph signal recovery. Finally, we illustrate algorithms and optimal sampling strategies for adaptive recovery and tracking of dynamic graph signals, where both sampling set and …
Recent progress in graph signal processing (GSP) has addressed a number of problems, including sampling and filtering. Proposed methods have focused on generic graphs and defined signals with certain characteristics, e.g., bandlimited signals, based on t he graph Fourier transform (GFT). However, the effect of GFT properties (e.g., vertex localization) on the behavior of such methods is not as well understood. In this paper, we propose novel GFT visualization tools and provide some examples to illustrate certain GFT properties and their impact on sampling or wavelet transforms.
