Motivated by the need to model joint dependence between regions of interest in functional neuroconnectivity for efficient inference, we propose a new sampling-based Bayesian clustering approach for covariance structures of high-dimensional Gaussian outcomes. The key technique is based on a Dirichlet process that clusters covariance sub-matrices into independent groups of outcomes, thereby naturally inducing sparsity in the whole brain connectivity matrix. A new split-merge algorithm is employed to improve the mixing of the Markov chain sampling that is shown empirically to recover both uniform and Dirichlet partitions with high accuracy. We investigate the empirical performance of the proposed method through extensive simulations. Finally, the proposed approach is used to group regions of interest into functionally independent groups in the Autism Brain Imaging Data Exchange participants with autism spectrum disorder and attention-deficit/hyperactivity disorder.
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This note is an attempt to unconditionally prove the existence of weak one way functions (OWF). Starting from a provably intractable decision problem $L_D$ (whose existence is nonconstructively assured from the well-known discrete time-hierarchy theorem from complexity theory), we construct another intractable decision problem $L\subseteq \{0,1\}^*$ that has its words scattered across $\{0,1\}^\ell$ at a relative frequency $p(\ell)$, for which upper and lower bounds can be worked out. The value $p(\ell)$ is computed from the density of the language within $\{0,1\}^\ell$ divided by the total word count $2^\ell$. It corresponds to the probability of retrieving a yes-instance of a decision problem upon a uniformly random draw from $\{0,1\}^\ell$. The trick to find a language with known bounds on $p(\ell)$ relies on switching from $L_D$ to $L_0:=L_D\cap L'$, where $L'$ is an easy-to-decide language with a known density across $\{0,1\}^*$. In defining $L'$ properly (and upon a suitable G\"odel numbering), the hardness of deciding $L_D\cap L'$ is inherited from $L_D$, while its density is controlled by that of $L'$. The lower and upper approximation of $p(\ell)$ then let us construct an explicit threshold function (as in random graph theory) that can be used to efficiently and intentionally sample yes- or no-instances of the decision problem (language) $L_0$ (however, without any auxiliary information that could ease the decision like a polynomial witness). In turn, this allows to construct a weak OWF that encodes a bit string $w\in\{0,1\}^*$ by efficiently (in polynomial time) emitting a sequence of randomly constructed intractable decision problems, whose answers correspond to the preimage $w$.
The target sensing/localization performance is fundamentally limited by the line-of-sight link and severe signal attenuation over long distances. This paper considers a challenging scenario where the direct link between the base station (BS) and the target is blocked due to the surrounding blockages and leverages the intelligent reflecting surface (IRS) with some active sensors, termed as \textit{semi-passive IRS}, for localization. To be specific, the active sensors receive echo signals reflected by the target and apply signal processing techniques to estimate the target location. We consider the joint time-of-arrival (ToA) and direction-of-arrival (DoA) estimation for localization and derive the corresponding Cram\'{e}r-Rao bound (CRB), and then a simple ToA/DoA estimator without iteration is proposed. In particular, the relationships of the CRB for ToA/DoA with the number of frames for IRS beam adjustments, number of IRS reflecting elements, and number of sensors are theoretically analyzed and demystified. Simulation results show that the proposed semi-passive IRS architecture provides sub-meter level positioning accuracy even over a long localization range from the BS to the target and also demonstrate a significant localization accuracy improvement compared to the fully passive IRS architecture.
In this paper, we focus on the high-dimensional double sparse structure, where the parameter of interest simultaneously encourages group-wise sparsity and element-wise sparsity in each group. By combining the Gilbert-Varshamov bound and its variants, we develop a novel lower bound technique for the metric entropy of the parameter space, specifically tailored for the double sparse structure over $\ell_u(\ell_q)$-balls with $u,q \in [0,1]$. We prove lower bounds on the estimation error using an information-theoretic approach, leveraging our proposed lower bound technique and Fano's inequality. To complement the lower bounds, we establish matching upper bounds through a direct analysis of constrained least-squares estimators and utilize results from empirical processes. A significant finding of our study is the discovery of a phase transition phenomenon in the minimax rates for $u,q \in (0, 1]$. Furthermore, we extend the theoretical results to the double sparse regression model and determine its minimax rate for estimation error. To tackle double sparse linear regression, we develop the DSIHT (Double Sparse Iterative Hard Thresholding) algorithm, demonstrating its optimality in the minimax sense. Finally, we demonstrate the superiority of our method through numerical experiments.
Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.
We propose a general purpose confidence interval procedure (CIP) for statistical functionals constructed using data from a stationary time series. The procedures we propose are based on derived distribution-free analogues of the $\chi^2$ and Student's $t$ random variables for the statistical functional context, and hence apply in a wide variety of settings including quantile estimation, gradient estimation, M-estimation, CVAR-estimation, and arrival process rate estimation, apart from more traditional statistical settings. Like the method of subsampling, we use overlapping batches of time series data to estimate the underlying variance parameter; unlike subsampling and the bootstrap, however, we assume that the implied point estimator of the statistical functional obeys a central limit theorem (CLT) to help identify the weak asymptotics (called OB-x limits, x=I,II,III) of batched Studentized statistics. The OB-x limits, certain functionals of the Wiener process parameterized by the size of the batches and the extent of their overlap, form the essential machinery for characterizing dependence, and consequently the correctness of the proposed CIPs. The message from extensive numerical experimentation is that in settings where a functional CLT on the point estimator is in effect, using \emph{large overlapping batches} alongside OB-x critical values yields confidence intervals that are often of significantly higher quality than those obtained from more generic methods like subsampling or the bootstrap. We illustrate using examples from CVaR estimation, ARMA parameter estimation, and NHPP rate estimation; R and MATLAB code for OB-x critical values is available at~\texttt{web.ics.purdue.edu/~pasupath/}.
We consider hypergraph network design problems where the goal is to construct a hypergraph satisfying certain properties. In graph network design problems, the number of edges in an arbitrary solution is at most the square of the number of vertices. In contrast, in hypergraph network design problems, the number of hyperedges in an arbitrary solution could be exponential in the number of vertices and hence, additional care is necessary to design polynomial-time algorithms. The central theme of this work is to show that certain hypergraph network design problems admit solutions with polynomial number of hyperedges and moreover, can be solved in strongly polynomial time. Our work improves on the previous fastest pseudo-polynomial run-time for these problems. In addition, we develop algorithms that return (near-)uniform hypergraphs as solutions. The hypergraph network design problems that we focus upon are splitting-off operation in hypergraphs, connectivity augmentation using hyperedges, and covering skew-supermodular functions using hyperedges. Our definition of the splitting-off operation in hypergraphs and our proof showing the existence of the operation using a strongly polynomial-time algorithm to compute it are likely to be of independent graph-theoretical interest.
We consider functional data where an underlying smooth curve is composed not just with errors, but also with irregular spikes. We propose an approach that, combining regularized spline smoothing and an Expectation-Maximization algorithm, allows one to both identify spikes and estimate the smooth component. Imposing some assumptions on the error distribution, we prove consistency of EM estimates. Next, we demonstrate the performance of our proposal on finite samples and its robustness to assumptions violations through simulations. Finally, we apply our proposal to data on the annual heatwaves index in the US and on weekly electricity consumption in Ireland. In both datasets, we are able to characterize underlying smooth trends and to pinpoint irregular/extreme behaviors.
Population size estimation based on the capture-recapture experiment is an interesting problem in various fields including epidemiology, criminology, demography, etc. In many real-life scenarios, there exists inherent heterogeneity among the individuals and dependency between capture and recapture attempts. A novel trivariate Bernoulli model is considered to incorporate these features, and the Bayesian estimation of the model parameters is suggested using data augmentation. Simulation results show robustness under model misspecification and the superiority of the performance of the proposed method over existing competitors. The method is applied to analyse real case studies on epidemiological surveillance. The results provide interesting insight on the heterogeneity and dependence involved in the capture-recapture mechanism. The methodology proposed can assist in effective decision-making and policy formulation.
The $k$-means algorithm is a prevalent clustering method due to its simplicity, effectiveness, and speed. However, its main disadvantage is its high sensitivity to the initial positions of the cluster centers. The global $k$-means is a deterministic algorithm proposed to tackle the random initialization problem of k-means but its well-known that requires high computational cost. It partitions the data to $K$ clusters by solving all $k$-means sub-problems incrementally for all $k=1,\ldots, K$. For each $k$ cluster problem, the method executes the $k$-means algorithm $N$ times, where $N$ is the number of datapoints. In this paper, we propose the \emph{global $k$-means\texttt{++}} clustering algorithm, which is an effective way of acquiring quality clustering solutions akin to those of global $k$-means with a reduced computational load. This is achieved by exploiting the center selection probability that is effectively used in the $k$-means\texttt{++} algorithm. The proposed method has been tested and compared in various benchmark datasets yielding very satisfactory results in terms of clustering quality and execution speed.
Off-policy evaluation (OPE) aims to estimate the benefit of following a counterfactual sequence of actions, given data collected from executed sequences. However, existing OPE estimators often exhibit high bias and high variance in problems involving large, combinatorial action spaces. We investigate how to mitigate this issue using factored action spaces i.e. expressing each action as a combination of independent sub-actions from smaller action spaces. This approach facilitates a finer-grained analysis of how actions differ in their effects. In this work, we propose a new family of "decomposed" importance sampling (IS) estimators based on factored action spaces. Given certain assumptions on the underlying problem structure, we prove that the decomposed IS estimators have less variance than their original non-decomposed versions, while preserving the property of zero bias. Through simulations, we empirically verify our theoretical results, probing the validity of various assumptions. Provided with a technique that can derive the action space factorisation for a given problem, our work shows that OPE can be improved "for free" by utilising this inherent problem structure.
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