This article is motivated by studying multisensory effects on brain activities in intracranial electroencephalography (iEEG) experiments. Differential brain activities to multisensory stimulus presentations are zero in most regions and non-zero in some local regions, yielding locally sparse functions. Such studies are essentially a function-on-scalar regression problem, with interest being focused not only on estimating nonparametric functions but also on recovering the function supports. We propose a weighted group bridge approach for simultaneous function estimation and support recovery in function-on-scalar mixed effect models, while accounting for heterogeneity present in functional data. We use B-splines to transform sparsity of functions to its sparse vector counterpart of increasing dimension, and propose a fast non-convex optimization algorithm using nested alternative direction method of multipliers (ADMM) for estimation. Large sample properties are established. In particular, we show that the estimated coefficient functions are rate optimal in the minimax sense under the $L_2$ norm and resemble a phase transition phenomenon. For support estimation, we derive a convergence rate under the $L_{\infty}$ norm that leads to a sparsistency property under $\delta$-sparsity, and provide a simple sufficient regularity condition under which a strict sparsistency property is established. An adjusted extended Bayesian information criterion is proposed for parameter tuning. The developed method is illustrated through simulation and an application to a novel iEEG dataset to study multisensory integration. We integrate the proposed method into RAVE, an R package that gains increasing popularity in the iEEG community.
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Matrix approximations are a key element in large-scale algebraic machine learning approaches. The recently proposed method MEKA (Si et al., 2014) effectively employs two common assumptions in Hilbert spaces: the low-rank property of an inner product matrix obtained from a shift-invariant kernel function and a data compactness hypothesis by means of an inherent block-cluster structure. In this work, we extend MEKA to be applicable not only for shift-invariant kernels but also for non-stationary kernels like polynomial kernels and an extreme learning kernel. We also address in detail how to handle non-positive semi-definite kernel functions within MEKA, either caused by the approximation itself or by the intentional use of general kernel functions. We present a Lanczos-based estimation of a spectrum shift to develop a stable positive semi-definite MEKA approximation, also usable in classical convex optimization frameworks. Furthermore, we support our findings with theoretical considerations and a variety of experiments on synthetic and real-world data.
This paper studies the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. Our most general result is a perturbation bound of the signal space in terms of the minimum singular value of Fourier matrices. When the point sources are located in several separated clumps, we provide an explicit upper bound of the noise-space correlation perturbation error in MUSIC and the support error in ESPRIT in terms of a Super-Resolution Factor (SRF). The upper bound for ESPRIT is then compared with a new Cram\'er-Rao lower bound for the clumps model. As a result, we show that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems. Our theory is validated by numerical experiments.
We model and study the problem of localizing a set of sparse forcing inputs for linear dynamical systems from noisy measurements when the initial state is unknown. This problem is of particular relevance to detecting forced oscillations in electric power networks. We express measurements as an additive model comprising the initial state and inputs grouped over time, both expanded in terms of the basis functions (i.e., impulse response coefficients). Using this model, with probabilistic guarantees, we recover the locations and simultaneously estimate the initial state and forcing inputs using a variant of the group LASSO (linear absolute shrinkage and selection operator) method. Specifically, we provide a tight upper bound on: (i) the probability that the group LASSO estimator wrongly identifies the source locations, and (ii) the $\ell_2$-norm of the estimation error. Our bounds explicitly depend upon the length of the measurement horizon, the noise statistics, the number of inputs and sensors, and the singular values of impulse response matrices. Our theoretical analysis is one of the first to provide a complete treatment for the group LASSO estimator for linear dynamical systems under input-to-output delay assumptions. Finally, we validate our results on synthetic models and the IEEE 68-bus, 16-machine system.
We prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. \"Osterreich. Akad. Wiss. Math.-Natur. Kl. 132 (1995), 3--10]. In particular, we see that the inverse $N(\varepsilon,d)$ of the minimal spherical dispersion is, for fixed $\varepsilon>0$, linear in the dimension $d$ of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse $\widetilde{N}(\varepsilon,d)$, our bounds are optimal with respect to the dependence on $\varepsilon$.
Extremely large-scale multiple-input-multiple-output (XL-MIMO) with hybrid precoding is a promising technique to meet the high data rate requirements for future 6G communications. To realize efficient hybrid precoding, it is essential to obtain accurate channel state information. Existing channel estimation algorithms with low pilot overhead heavily rely on the channel sparsity in the angle domain, which is achieved by the classical far-field planar wavefront assumption. However, due to the non-negligible near-field spherical wavefront property in XL-MIMO systems, this channel sparsity in the angle domain is not available anymore, and thus existing far-field channel estimation schemes will suffer from severe performance loss. To address this problem, in this paper we study the near-field channel estimation by exploiting the polar-domain sparse representation of the near-field XL-MIMO channel. Specifically, unlike the classical angle-domain representation that only considers the angle information of the channel, we propose a new polar-domain representation, which simultaneously accounts for both the angle and distance information. In this way, the near-field channel also exhibits sparsity in the polar domain. By exploiting the channel sparsity in the polar domain, we propose the on-grid and off-grid near-field channel estimation schemes for XL-MIMO. Firstly, an on-grid polar-domain simultaneous orthogonal matching pursuit (P-SOMP) algorithm is proposed to efficiently estimate the near-field channel. Furthermore, to solve the resolution limitation of the on-grid P-SOMP algorithm, an off-grid polar-domain simultaneous iterative gridless weighted (P-SIGW) algorithm is proposed to improve the estimation accuracy, where the parameters of the near-field channel are directly estimated. Finally, numerical results are provided to verify the effectiveness of the proposed schemes.
We show that solution to the Hermite-Pad\'{e} type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev-Petviashvili) system and of its adjoint linear problem. Our result explains the appearence of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorthms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite-Pad\'{e} approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite-Pad\'{e} problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.
In this paper, we present three estimators of the ROC curve when missing observations arise among the biomarkers. Two of the procedures assume that we have covariates that allow to estimate the propensity and the estimators are obtained using an inverse probability weighting method or a smoothed version of it. The other one assumes that the covariates are related to the biomarkers through a regression model which enables us to construct convolution--based estimators of the distribution and quantile functions. Consistency results are obtained under mild conditions. Through a numerical study we evaluate the finite sample performance of the different proposals. A real data set is also analysed.
In this paper, we study a non-local approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the non-local problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and non-local problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the non-local problem and that of the local one, both in continuous-time and Backward Euler time discretization. We then turn to studying continuum limits of non-local problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows, then almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.
We propose a joint channel estimation and signal detection approach for the uplink non-orthogonal multiple access using unsupervised machine learning. We apply the Gaussian mixture model to cluster the received signals, and accordingly optimize the decision regions to enhance the symbol error rate (SER). We show that, when the received powers of the users are sufficiently different, the proposed clustering-based approach achieves an SER performance on a par with that of the conventional maximum-likelihood detector with full channel state information. However, unlike the proposed approach, the maximum-likelihood detector requires the transmission of a large number of pilot symbols to accurately estimate the channel. The accuracy of the utilized clustering algorithm depends on the number of the data points available at the receiver. Therefore, there exists a tradeoff between accuracy and block length. We provide a comprehensive performance analysis of the proposed approach as well as deriving a theoretical bound on its SER performance as a function of the block length. Our simulation results corroborate the effectiveness of the proposed approach and verify that the calculated theoretical bound can predict the SER performance of the proposed approach well.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
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