State Evolution Analysis Research Articles

A Copula-based framework for joint encounter and state evolution analysis of urban short-duration rainstorm characteristics

In this study, in order to reveal the joint encounter and state evolution characteristics of urban short-duration rainstorms, a case study is conducted in Zhengzhou city, China. The joint probability distribution model of total rainfall (P) and peak rainfall (R) is constructed by using the Copula function, and their joint encounter and joint return period are analyzed. Additionally, the state evolution characteristics of P and R are explored by incorporating with the Markov chain. The results show that: (1) there is a good correlation between P and R under different rainfall durations. The Clayton Copula function is found suitable for 0.5 h and 1 h rainfall durations, while the Frank Copula function is selected for the 1.5 h rainfall duration. The P and R are more densely distributed in the joint probability interval of 0.2–0.7 and the joint return period of rainfall events (JRPRE) of 15-30freq, which are particularly sensitive to the changes in the corresponding intervals. (2) The probability of synchronous rich (F)-low (L) between P and R is greater than that of asynchronous F-L. The probability of synchronous F-L increases with longer rainfall durations, reaching the highest value of 74.92 % for the 1.5 h rainfall duration. (3) The transition probability and limit probability of the F/high (H) state and F-H combination state are relatively large, indicating that P and R exhibit strong self-conservation. These states are associated with the highest risk and tend to occur more frequently with shorter average return periods of rainfall events (ARPRE). The findings are valuable in uncovering the occurrence patterns of short-duration rainstorms and facilitating measures to mitigate waterlogging, thereby supporting urban sustainable development.

Fundamental limits in structured principal component analysis and how to reach them

How do statistical dependencies in measurement noise influence high-dimensional inference? To answer this, we study the paradigmatic spiked matrix model of principal components analysis (PCA), where a rank-one matrix is corrupted by additive noise. We go beyond the usual independence assumption on the noise entries, by drawing the noise from a low-order polynomial orthogonal matrix ensemble. The resulting noise correlations make the setting relevant for applications but analytically challenging. We provide characterization of the Bayes optimal limits of inference in this model. If the spike is rotation invariant, we show that standard spectral PCA is optimal. However, for more general priors, both PCA and the existing approximate message-passing algorithm (AMP) fall short of achieving the information-theoretic limits, which we compute using the replica method from statistical physics. We thus propose an AMP, inspired by the theory of adaptive Thouless-Anderson-Palmer equations, which is empirically observed to saturate the conjectured theoretical limit. This AMP comes with a rigorous state evolution analysis tracking its performance. Although we focus on specific noise distributions, our methodology can be generalized to a wide class of trace matrix ensembles at the cost of more involved expressions. Finally, despite the seemingly strong assumption of rotation-invariant noise, our theory empirically predicts algorithmic performance on real data, pointing at strong universality properties.

Rigorous State Evolution Analysis for Approximate Message Passing With Side Information

A common goal in many research areas is to reconstruct an unknown signal x from noisy linear measurements. Approximate message passing (AMP) is a class of low-complexity algorithms that can be used for efficiently solving such high-dimensional regression tasks. Often, it is the case that side information (SI) is available during reconstruction, for example in online learning applications. For this reason, a novel algorithmic framework that incorporates SI into AMP, referred to as approximate message passing with side information (AMP-SI), has been recently introduced. In this work, we provide rigorous performance guarantees for AMP-SI when there are statistical dependencies between the signal and SI pairs and the entries of the measurement matrix are independent and identically distributed (i.i.d.) Gaussian. We also allow for statistical dependencies within the elements of the signal itself, by considering a flexible AMP-SI framework incorporating both separable and non-separable denoisers. The AMP-SI performance is shown to be provably tracked by a scalar iteration referred to as state evolution (SE). Moreover, we provide numerical examples that demonstrate empirically that the SE can predict the AMP-SI mean square error accurately.

Massive MIMO Precoding and Spectral Shaping With Low Resolution Phase-Only DACs and Active Constellation Extension

Nonlinear precoding and pulse shaping are jointly considered in multi-user massive multiple-input multiple-output (MIMO) systems with low resolution D/A-converters (DACs) in terms of algorithmic approach as well as large system performance. Two design criteria are investigated: the mean squared error (MSE) with active constellation extension (ACE) and the symbol error rate (SER). Both formulations are solved based on a modified version of the generalized approximate message passing (GAMP) algorithm. Furthermore, theoretical performance results are derived based on the state evolution analysis of the GAMP algorithm. The MSE based technique is extended to jointly perform over-the-air (OTA) spectral shaping and precoding for frequency-selective channels, in which the spectral performance is characterized at the transmitter and at the receiver. Simulation and analytical results demonstrate that the MSE based approach yields the same performance as the SER based formulation in terms of uncoded SER. The analytical results provide good performance predictions up to medium SNR. Substantial improvements in detection, as well as spectral performance, are obtained from the proposed combined pulse shaping and precoding approach compared to standard linear methods.

Low-Complexity Coding and Spreading for Unsourced Random Access

We propose a low-complexity unsourced random access (URA) transmission scheme where the data payload, encoded by an error-reduction code, undergoes repetition, permutation, and spreading. An approach to perform spatial graph coupling of the URA messages via randomized message delays is also presented. The proposed system, in either block or coupled form, outperforms the existing URA counterparts in the high multi-user interference regime and provides a substantial increase in the supported multi-user loads. A choice of stronger, albeit more complex, error-correction code allows for performance improvements at lower system loads. Finally, system performance predictions via a state-evolution analysis are also demonstrated.

Compressed Coding, AMP-Based Decoding, and Analog Spatial Coupling

This paper considers a compressed-coding scheme that combines compressed sensing with forward error control coding. Approximate message passing (AMP) is used to decode the message. Based on the state evolution analysis of AMP, we derive the performance limit of compressed-coding. We show that compressed-coding can approach Gaussian capacity at a very low compression ratio. Further, the results are extended to systems involving non-linear effects such as clipping. We show that the capacity approaching property can still be maintained when generalized AMP is used to decode the message. To approach the capacity, a low-rate underlying code should be designed according to the curve matching principle, which is complicated in practice. Instead, analog spatial-coupling is used to avoid sophisticated low-rate code design. In the end, we study the coupled scheme in a multiuser environment, where analog spatial-coupling can be realized in a distributive way. The overall block length can be shared by many users, which reduces block length per-user.

Mutual Information and Optimality of Approximate Message-Passing in Random Linear Estimation

We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections. A few examples where this problem is relevant are compressed sensing, sparse superposition codes, and code division multiple access. There has been a number of works considering the mutual information for this problem using the replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-Toninelli type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical cases, we present a converse lower bound via a method that uses spatial coupling, state evolution analysis and the I-MMSE theorem. This yields a single letter formula for the mutual information and the minimal-mean-square error for random Gaussian linear estimation of all discrete bounded signals. In addition, we prove that the low complexity approximate message-passing algorithm is optimal outside of the so-called hard phase, in the sense that it asymptotically reaches the minimal-mean-square error. In this work spatial coupling is used primarily as a proof technique. However our results also prove two important features of spatially coupled noisy linear random Gaussian estimation. First there is no algorithmically hard phase. This means that for such systems approximate message-passing always reaches the minimal-mean-square error. Secondly, in a proper limit the mutual information associated to such systems is the same as the one of uncoupled linear random Gaussian estimation.

Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing

SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this paper, we develop an asymptotically exact characterization of the SLOPE solution under Gaussian random designs through solving the SLOPE problem using approximate message passing (AMP). This algorithmic approach allows us to approximate the SLOPE solution via the much more amenable AMP iterates. Explicitly, we characterize the asymptotic dynamics of the AMP iterates relying on a recently developed state evolution analysis for non-separable penalties, thereby overcoming the difficulty caused by the sorted ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> penalty. Moreover, we prove that the AMP iterates converge to the SLOPE solution in an asymptotic sense, and numerical simulations show that the convergence is surprisingly fast. Our proof rests on a novel technique that specifically leverages the SLOPE problem. In contrast to prior literature, our work not only yields an asymptotically sharp analysis but also offers an algorithmic, flexible, and constructive approach to understanding the SLOPE problem.

Universal Sparse Superposition Codes With Spatial Coupling and GAMP Decoding

Sparse superposition codes, or sparse regression codes, constitute a new class of codes, which was first introduced for communication over the additive white Gaussian noise (AWGN) channel. It has been shown that such codes are capacity-achieving over the AWGN channel under optimal maximum-likelihood decoding as well as under various efficient iterative decoding schemes equipped with power allocation or spatially coupled constructions. Here, we generalize the analysis of these codes to a much broader setting that includes all memoryless channels. We show, for a large class of memoryless channels, that spatial coupling allows an efficient decoder, based on the generalized approximate message-passing (GAMP) algorithm, to reach the potential (or Bayes optimal) threshold of the underlying (or uncoupled) code ensemble. Moreover, we argue that spatially coupled sparse superposition codes universally achieve capacity under GAMP decoding by showing, through analytical computations, that the error floor vanishes and the potential threshold tends to capacity, as one of the code parameters goes to infinity. Furthermore, we provide a closed-form formula for the algorithmic threshold of the underlying code ensemble in terms of Fisher information. Relating an algorithmic threshold to a Fisher information has theoretical as well as practical importance. Our proof relies on the state evolution analysis and uses the potential method developed in the theory of low-density parity-check (LDPC) codes and compressed sensing.

Structured Turbo Compressed Sensing for Massive MIMO Channel Estimation Using a Markov Prior

Accurate channel estimation with small pilot overhead is vital to improve the capacity and reliability of massive MIMO systems. Recently, compressed sensing has been applied to reduce the pilot overhead in such systems by exploiting the underlying structured channel sparsity. In this paper, we propose a structured turbo compressed sensing (Turbo-CS) framework for the design and analysis of structured sparse channel estimation algorithms. In this framework, a Markov prior is used to model the structured sparsity in massive MIMO channels. Then we extend the Turbo-CS algorithm for independent and identically distributed priors to propose a structured Turbo-CS algorithm to solve the resulting sparse channel estimation problem with the Markov chain prior. We also accurately characterize the performance of the algorithm using state evolution. As compared to the existing algorithms, both the state evolution analysis and simulations show that the structured Turbo-CS algorithm can substantially enhance the channel estimation performance.

One-Bit Quantized Massive MIMO Detection Based on Variational Approximate Message Passing

One-bit quantization can significantly reduce the massive multiple-input and multiple-output (MIMO) system hardware complexity, but at the same time it also brings great challenges to the system algorithm design. Specifically, it is difficult to recover information from the highly distorted samples as well as to obtain accurate channel estimation without increasing the number of pilots. In this paper, a novel inference algorithm called variational approximate message passing (VAMP) for one-bit quantized massive MIMO receiver is developed, which attempts to exploit the advantages of both the variational Bayesian inference algorithm and the bilinear generalized approximated message passing algorithm to accomplish joint channel estimation and data detection in a closed form with first-order complexity. Asymptotic state evolution analysis indicates the fast convergence rate of VAMP and also provides a lower bound for the data detection error. Moreover, through extensive simulations, we show that VAMP can achieve excellent detection performance with low pilot overhead in a wide range of scenarios.

Phase Transitions and Sample Complexity in Bayes-Optimal Matrix Factorization

We analyse the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics - the cavity and replica methods - to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable in principle in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error, and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.

On the Performance of Turbo Signal Recovery with Partial DFT Sensing Matrices

This letter is on the performance of the turbo signal recovery (TSR) algorithm for partial discrete Fourier transform (DFT) matrices based compressed sensing. Based on state evolution analysis, we prove that TSR with a partial DFT sensing matrix outperforms the well-known approximate message passing (AMP) algorithm with an independent identically distributed (IID) sensing matrix.

Urban Regional Traffic State Analysis Software System Emphasizing Pattern Transition

Urban traffic state evolution analysis is very significant and constructive for traffic guidance and control. In this paper, firstly, a quantitative method for analyzing regional traffic state evolution was proposed by constructing traffic state pattern transition network to mine regional traffic state information and state pattern transition characteristics from massive data. Secondly, a GIS-based urban regional traffic state analysis soft ware system URTSAS based on the method was designed and implemented. Application scenarios and effect shows that traffic state transition analysis in URTSAS has a wide application prospect in improving urban traffic management and decision-making.

Dynamics of high repetition rate regenerative amplifiers

Dynamics features of high repetition rate continuously pumped solid-state regenerative amplifiers were studied numerically. A space independent rate equations and discrete-time dynamical system approach were used for system state evolution analysis. Regular single-energy operation, quasi-periodic pulsing and chaotic behavior regions are distinguished in space of control parameters. Diagrams of dynamical regimes comprehensively exhibiting operation features of the system are presented. Seed energy is shown to be an important parameter determining the stability space. Conditions of stable operation are described quantitatively.