Finite Number Of Samples Research Articles

Sampling Error Analysis in Quantum Krylov Subspace Diagonalization

Quantum Krylov subspace diagonalization (QKSD) is an emerging method used in place of quantum phase estimation in the early fault-tolerant era, where limited quantum circuit depth is available. In contrast to the classical Krylov subspace diagonalization (KSD) or the Lanczos method, QKSD exploits the quantum computer to efficiently estimate the eigenvalues of large-size Hamiltonians through a faster Krylov projection. However, unlike classical KSD, which is solely concerned with machine precision, QKSD is inherently accompanied by errors originating from a finite number of samples. Moreover, due to difficulty establishing an artificial orthogonal basis, ill-conditioning problems are often encountered, rendering the solution vulnerable to noise. In this work, we present a nonasymptotic theoretical framework to assess the relationship between sampling noise and its effects on eigenvalues. We also propose an optimal solution to cope with large condition numbers by eliminating the ill-conditioned bases. Numerical simulations of the one-dimensional Hubbard model demonstrate that the error bound of finite samplings accurately predicts the experimental errors in well-conditioned regions.

DEA에 의한 전통시장의 효율성측정 개선에 관한 연구

In order to increase the effectiveness of traditional market policies, in which enormous funds are invested, it can be said that objective and transparent performance evaluation of traditional markets takes precedence. This study proposed two improvements to the existing DEA method in order to improve efficiency studies on traditional markets. In other words, by using a two-step approach using bootstrap, we first calculated the bias-corrected efficiency and examined how external environmental factors affect the efficiency of the traditional market. First, in order to improve the current DEA efficiency index, which is overestimated based on a finite number of samples, we tried to obtain a bias-corrected efficiency index through bootstrapping. This improvement has caused significant changes in the efficiency rankings among existing traditional markets. Second, truncated regression analysis was conducted by bootstrapping to explain the differences in efficiency between traditional markets that inevitably occur depending on the external environment such as location of traditional markets. As a result, it was found that the existence of large retail stores had a negative effect on the efficiency of traditional markets, while the existence of public facilities and transportation facilities had a positive effect on efficiency. If such an external environment becomes an important cause for the difference in efficiency between traditional markets, regardless of internal management efforts in traditional markets, it can be said that appropriate adjustments and modifications are required in performance evaluation.

Sampling and Reconstructing Angular Domains With Uniform Arrays

The surge of massive antenna arrays in wireless networks calls for the adoption of analog/hybrid array solutions, where multiple antenna elements are driven by a common radio front end to form a beam along a specific angle in order to maximize the beamforming gain. Many heuristics have been proposed to sample the angular domain by trading off between sampling step size and overhead, where arbitrarily small angular step size is only attainable with infinite sampling overhead. We show that, for uniform linear and rectangular arrays, lossless reconstruction of the array’s angular responses at arbitrary angular precision is possible using a finite number of samples without resorting to assumptions of angular sparsity. The proposed method, sampling and reconstructing angular domain (SARA), defines how many and which angles to be sampled and the corresponding reconstruction. This general solution to scan the angular domain can therefore be applied not only to beam acquisition and channel estimation, but also to radio imaging techniques, making it a candidate for future integrated sensing and communications (ISAC). Extensive simulation results for target detection and radio imaging have demonstrated clear advantages of SARA over other considered baselines, both in terms of angular reconstruction performance and computational complexity.

Safe and Efficient Dynamic Window Approach for Differential Mobile Robots With Stochastic Dynamics Using Deterministic Sampling

We propose an efficient and safe dynamic window approach (DWA) by using deterministic sampling. When the system dynamics have uncertainty, the control input includes errors, so that the DWA objective function becomes a random variable. When a random-choice algorithm with a finite number of samples is used to estimate the objective function, it may miss collisions during prediction. In this work, we approximate the end-state distribution as a one-dimensional distribution for each input candidate in advance and generate sample paths deterministically to eliminate the misses to achieve safe control. Numerical experiments have shown that this method is approximately three times as efficient as the Monte Carlo method in most indoor environments.

Quantification of limitations in statistical analysis of ultrasound echo envelope amplitudes

Ultrasound echo envelope statistics have been widely studied for quantitative tissue characterization. In ultrasound measurements, the size of the region-of-interest (ROI) is limited by several factors, such as the locality of the tissue characteristics and the depth dependence of the acoustic field of the ultrasound beam. In this case, the evaluated echo envelope statistics vary even when the envelope amplitudes follow the same population without any noise. In this study, the statistical variance of the moments caused by this finite number of samples was quantified based on the central limit theorem and the law of error propagation. The proposed principles were validated by random number simulation and used to quantify the statistical variance of Nakagami parameter estimation. Finally, the effective number of independent samples in an ultrasonic measurement was quantified based on the relationship between the ROI size and the ultrasound spatial resolution.

A robust estimator of mutual information for deep learning interpretability

We develop the use of mutual information (MI), a well-established metric in information theory, to interpret the inner workings of deep learning (DL) models. To accurately estimate MI from a finite number of samples, we present GMM-MI (pronounced ‘Jimmie’), an algorithm based on Gaussian mixture models that can be applied to both discrete and continuous settings. GMM-MI is computationally efficient, robust to the choice of hyperparameters and provides the uncertainty on the MI estimate due to the finite sample size. We extensively validate GMM-MI on toy data for which the ground truth MI is known, comparing its performance against established MI estimators. We then demonstrate the use of our MI estimator in the context of representation learning, working with synthetic data and physical datasets describing highly non-linear processes. We train DL models to encode high-dimensional data within a meaningful compressed (latent) representation, and use GMM-MI to quantify both the level of disentanglement between the latent variables, and their association with relevant physical quantities, thus unlocking the interpretability of the latent representation. We make GMM-MI publicly available in this GitHub repository.

Data-Driven Abstractions With Probabilistic Guarantees for Linear PETC Systems

We employ the scenario approach to compute probably approximately correct (PAC) bounds on the average inter-sample time (AIST) generated by an unknown PETC system, based on a finite number of samples. We extend the scenario optimisation to multiclass SVM algorithms in order to construct a PAC map between the concrete state-space and the inter-sample times. We then build a traffic model applying an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -complete relation and find, in the underlying graph, the cycles of minimum and maximum average weight: these provide lower and upper bounds on the AIST. Numerical benchmarks show the practical applicability of our method, which is compared against model-based state-of-the-art tools.

Appropriate reduction of the posterior distribution in fully Bayesian inversions

SummaryBayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The prior is also introduced for the hyperparameters in fully Bayesian inversions and enables us to evaluate both the model parameters and hyperparameters probabilistically by the joint posterior. However, even in a linear inverse problem, it is unsolved how we should extract useful information on the model parameters from the joint posterior. This study presents a theoretical exploration into the appropriate dimensionality reduction of the joint posterior in the fully Bayesian inversion. We classify the ways of probability reduction into the following three categories focused on the marginalization of the joint posterior: (1) using the joint posterior without marginalization, (2) using the marginal posterior of the model parameters and (3) using the marginal posterior of the hyperparameters. First, we derive several analytical results that characterize these categories. One is a suite of semi-analytic representations of the probability maximization estimators for respective categories in the linear inverse problem. The mode estimators of categories (1) and (2) are found asymptotically identical for a large number of data and model parameters. We also prove the asymptotic distributions of categories (2) and (3) delta-functionally concentrate on their probability peaks, which predicts two distinct optimal estimates of the model parameters. Secondly, we conduct a synthetic test and find an appropriate reduction is realized by category (3), typified by Akaike’s Bayesian information criterion. The other reduction categories are shown inappropriate for the case of many model parameters, where the probability concentration of the marginal posterior of the model parameters no longer implies the central limit theorem. The main cause of these results is that the joint posterior peaks sharply at an underfitted or overfitted solution as the number of model parameters increases. The exponential growth of the probability space in the model-parameter dimension makes almost-zero-probability events finitely contribute to the posterior mean and distributions of categories (1) and (2) be pathological. One remedy for this pathology is counting all model-parameter realizations by integrating the joint posterior over the model-parameter space of exponential multiplicity. Hence, the marginal posterior of the hyperparameters for categories (3) becomes appropriate and can conform to the law of large numbers even with numerous model parameters. The exponential rarity of the posterior mean and ABIC estimates implies the exponential time complexity of ordinary Monte Carlo methods in population mean and ABIC computations. We also present a geophysical application to estimate a continuous strain-rate field from spatially discrete global navigation satellite system data, demonstrating denser basis function expansions of the model-parameter field lead to oversmoothed estimates in naive fully Bayesian approaches, while detailed fields are resolved with convergence by the reduction of category (3). We often naively believe a good solution can be constructed from a finite number of samples with high probabilities, but the high-probability domain could be inappropriate, and exponentially many samples become necessary for generating appropriate estimates in the high-dimensional fully Bayesian posterior probability space.

Regularized sampling reconstruction of signals in the linear canonical transform domain

Sampling reconstruction of signals band-limited in the linear canonical transform domain has become a major task in bridging the relationship between the continuous physical signals and discrete signals. Many recent works dealt with this task from different theoretical and practical aspects. Furthermore, and as in the case of the classical Shannon’s sampling theorem, attempts are made to accelerate the convergence sampling rate. In this work we incorporate two different regularization kernels in the sampling reconstruction of band-limited signals to remarkably enhance the convergence rate. The first kernel is a normalized Gaussian kernel, allowing the sampling reconstruction for not necessarily band-limited signals, while the second is a band-limited kernel. In the later case the bandwidth is rigorously considered to guarantee exact reconstruction. We estimate the truncation error since only a finite number of samples is available. Rigorous analytic proofs are provided with precise error bounds. Illustrative examples are carried out with comparisons with the classical reconstruction technique.

Discrete fractional Hankel transform based on a nonsymmetric kernel matrix

A discrete fractional Hankel transform (DFRHT) is developed based on the eigen decomposition of a nonsymmetric involutory kernel matrix Y of a discrete Hankel transform (DHT). The two eigen spaces pertaining to the two distinct eigenvalues of Y are not orthogonal because Y is not a normal matrix. Explicit expressions are derived for the oblique projection matrices of Y on its eigen spaces by applying a contributed generalized spectral theorem of linear algebra. Expressions are derived for the dimensions of the two eigen spaces in terms of the trace of matrix Y. An analytic form is derived for the inverse of a modal matrix of Y in order to circumvent the need for numerically evaluating it when computing the sophisticated fractional transform.The fact that the Hankel transform (HT) is self-reciprocating implies that taking the HT once can be viewed as a rotation by an angle π radians in the space – Hankel frequency plane. Being a generalization of the HT, the fractional Hankel transform (FRHT) of fractional order a corresponds to a rotation by an arbitrary angle α where α=πa.Initial orthonormal bases are individually generated for the two eigen spaces by the singular value decomposition of the oblique projection matrices on those spaces. In order for the DFRHT to approximate its continuous counterpart, namely the FRHT, it is desirable to define it in terms of eigenvectors which are as close as possible to samples of the eigen functions of the FRHT. After deriving a three term recursion formula for the eigen functions, a sampling scheme is proposed such that a finite number of samples will be representative of the behavior of those functions. Final optimal orthonormal eigenvectors are individually generated for the two eigen spaces.

A Forward-Backward Iterative Procedure for Improving the Resolution of Resonant Microwave Sensors

This paper sets out a method for improving the resolution of resonant microwave sensors. Usually, the frequency response of these devices is associated with a low quality factor, and consequently with a low resolution in terms of tracking capacity of the resonance frequency shift. Furthermore, since only a finite number of samples can be acquired during the measurement process, the “true” resonance frequency may not be included in the set of acquired data. In order to have an accurate estimate of the resonance frequency, high performance systems with very fine frequency sampling are thus required. To limit these drawbacks, an iterative algorithm is presented which aims to refine the response of resonant microwave sensors by means of a suitable post-processing. The algorithm evaluation is first carried out on synthetic data, and then applied on experimental data referring to a practical scenario, which is inherent to return loss measurements performed by a microwave patch antenna immersed in a water-glucose solution with different concentrations.

Deep Bayesian Quadrature Policy Optimization

We study the problem of obtaining accurate policy gradient estimates using a finite number of samples. Monte-Carlo methods have been the default choice for policy gradient estimation, despite suffering from high variance in the gradient estimates. On the other hand, more sample efficient alternatives like Bayesian quadrature methods have received little attention due to their high computational complexity. In this work, we propose deep Bayesian quadrature policy gradient (DBQPG), a computationally efficient high-dimensional generalization of Bayesian quadrature, for policy gradient estimation. We show that DBQPG can substitute Monte-Carlo estimation in policy gradient methods, and demonstrate its effectiveness on a set of continuous control benchmarks. In comparison to Monte-Carlo estimation, DBQPG provides (i) more accurate gradient estimates with a significantly lower variance, (ii) a consistent improvement in the sample complexity and average return for several deep policy gradient algorithms, and, (iii) the uncertainty in gradient estimation that can be incorporated to further improve the performance.

Efficient Sparse Coding Using Hierarchical Riemannian Pursuit

Sparse coding is a class of unsupervised methods for learning a sparse representation of the input data in the form of a linear combination of a dictionary and a sparse code. This learning framework has led to state-of-the-art results in various image and video processing tasks. However, classical methods learn the dictionary and the sparse code based on alternating optimizations, usually without theoretical guarantees for either optimality or convergence due to non-convexity of the problem. Recent works on sparse coding with a complete dictionary provide strong theoretical guarantees thanks to the development of the non-convex optimization. However, initial non-convex approaches learn the dictionary in the sparse coding problem sequentially in an atom-by-atom manner, which leads to a long execution time. More recent works seek to directly learn the entire dictionary at once, which substantially reduces the execution time. However, the associated recovery performance is degraded with a finite number of data samples. In this paper, we propose an efficient sparse coding scheme with a two-stage optimization. The proposed scheme leverages the global and local Riemannian geometry of the two-stage optimization problem and facilitates fast implementation for superb dictionary recovery performance by a finite number of samples without atom-by-atom calculation. We further prove that, with high probability, the proposed scheme can exactly recover any atom in the target dictionary with a finite number of samples if it is adopted to recover one atom of the dictionary. An application on wireless sensor data compression is also proposed. Experiments on both synthetic and real-world data verify the efficiency and effectiveness of the proposed scheme.

Wirtinger Flow Meets Constant Modulus Algorithm: Revisiting Signal Recovery for Grant-Free Access

In this work, we analyze the convergence of constant modulus algorithm (CMA) in blindly recovering multiple signals to facilitate grant-free wireless access. The CMA typically solves a non-convex problem by utilizing stochastic gradient descent. The iterative convergence of CMA can be affected by additive channel noise and finite number of samples, which is a problem not fully investigated previously. We point out the strong similarity between CMA and the Wirtinger Flow (WF) algorithm originally proposed for Phase retrieval. In light of the convergence proof of WF under limited data samples, we adopt the WF algorithm to implement CMA-based blind signal recovery. We generalize the convergence analysis of WF in the context of CMA-based blind signal recovery. Numerical simulation results also corroborate the analysis.

State-of-the-Art and Comparative Review of Adaptive Sampling Methods for Kriging

Metamodels aim to approximate characteristics of functions or systems from the knowledge extracted on only a finite number of samples. In recent years kriging has emerged as a widely applied metamodeling technique for resource-intensive computational experiments. However its prediction quality is highly dependent on the size and distribution of the given training points. Hence, in order to build proficient kriging models with as few samples as possible adaptive sampling strategies have gained considerable attention. These techniques aim to find pertinent points in an iterative manner based on information extracted from the current metamodel. A review of adaptive schemes for kriging proposed in the literature is presented in this article. The objective is to provide the reader with an overview of the main principles of adaptive techniques, and insightful details to pertinently employ available tools depending on the application at hand. In this context commonly applied strategies are compared with regards to their characteristics and approximation capabilities. In light of these experiments, it is found that the success of a scheme depends on the features of a specific problem and the goal of the analysis. In order to facilitate the entry into adaptive sampling a guide is provided. All experiments described herein are replicable using a provided open source toolbox.

Nonlinear Canonical Correlation Analysis:A Compressed Representation Approach.

Canonical Correlation Analysis (CCA) is a linear representation learning method that seeks maximally correlated variables in multi-view data. Nonlinear CCA extends this notion to a broader family of transformations, which are more powerful in many real-world applications. Given the joint probability, the Alternating Conditional Expectation (ACE) algorithm provides an optimal solution to the nonlinear CCA problem. However, it suffers from limited performance and an increasing computational burden when only a finite number of samples is available. In this work, we introduce an information-theoretic compressed representation framework for the nonlinear CCA problem (CRCCA), which extends the classical ACE approach. Our suggested framework seeks compact representations of the data that allow a maximal level of correlation. This way, we control the trade-off between the flexibility and the complexity of the model. CRCCA provides theoretical bounds and optimality conditions, as we establish fundamental connections to rate-distortion theory, the information bottleneck and remote source coding. In addition, it allows a soft dimensionality reduction, as the compression level is determined by the mutual information between the original noisy data and the extracted signals. Finally, we introduce a simple implementation of the CRCCA framework, based on lattice quantization.

Wasserstein distributionally robust shortest path problem

This paper proposes a data-driven distributionally robust shortest path (DRSP) model where the distribution of the travel time in the transportation network can only be partially observed through a finite number of samples. Specifically, we aim to find an optimal path to minimize the worst-case α-reliable mean-excess travel time (METT) over a Wasserstein ball, which is centered at the empirical distribution of the sample dataset and the ball radius quantifies the level of its confidence. In sharp contrast to the existing DRSP models, our model is equivalently reformulated as a tractable mixed 0–1 convex problem, e.g., 0–1 linear program or 0–1 second-order cone program. Moreover, we also explicitly derive the distribution achieving the worst-case METT by simply perturbing each sample. Experiments demonstrate the advantages of our DRSP model in terms of the out-of-sample performance and computational complexity. Finally, our DRSP model is easily extended to solve the distributionally robust bi-criteria shortest path problem and the minimum cost flow problem.

Distributionally Robust Fault Detection by using Kernel Density Estimation

In this paper, a method of distributionally robust fault detection (FD) is proposed for stochastic linear discrete-time systems by using the kernel density estimation (KDE) technique. For this purpose, an H2 optimization-based fault detection filter is constructed for residual generation. Towards maximizing the fault detection rate (FDR) for a prescribed false alarm rate (FAR), the residual evaluation issue regarding the design of residual evaluation function and threshold is formulated as a distributionally robust optimization problem, wherein the so-called confidence sets are constituted to model the ambiguity of distribution knowledge of residuals in fault-free and faulty cases. A KDE based solution, robust to the estimation errors in probability distribution of residual caused by the finite number of samples, is further developed to address the targeting problem such that the residual evaluation function, threshold as well as the lower bound of FDR can be achieved simultaneously. A case study on a vehicle lateral control system demonstrates the applicability of the proposed FD method.

Enhancing piecewise‐defined surrogate response surfaces with adjoints on sets of unstructured samples to solve stochastic inverse problems

SummaryMany approaches for solving stochastic inverse problems suffer from both stochastic and deterministic sources of error. The finite number of samples used to construct a solution is a common source of stochastic error. When computational models are expensive to evaluate, surrogate response surfaces are often employed to increase the number of samples available for approximating the solution. This leads to a reduction in finite sampling errors while the deterministic error in the evaluation of each sample is potentially increased. The pointwise accuracy of sampling the surrogate is primarily impacted by two sources of deterministic error: the local order of accuracy in the surrogate and the numerical error from the numerical solution of the model. In this work, we use adjoints to simultaneously give a posteriori error and derivative estimates in order to construct low‐order, piecewise‐defined surrogates on sets of unstructured samples. Several examples demonstrate the computational gains of this approach in obtaining accurate estimates of probabilities for events in the design space of model input parameters. This lays the groundwork for future studies on goal‐oriented adaptive refinement of such surrogates.

Learning a Local Hamiltonian from Local Measurements.

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.