Argmin Research Articles

Information Bottleneck as Optimisation Method for SSVEP-Based BCI.

In this study, the information bottleneck method is proposed as an optimisation method for steady-state visual evoked potential (SSVEP)-based brain-computer interface (BCI). The information bottleneck is an information-theoretic optimisation method for solving problems with a trade-off between preserving meaningful information and compression. Its main practical application in machine learning is in representation learning or feature extraction. In this study, we use the information bottleneck to find optimal classification rule for a BCI. This is a novel application for the information bottleneck. This approach is particularly suitable for BCIs since the information bottleneck optimises the amount of information transferred by the BCI. Steady-state visual evoked potential-based BCIs often classify targets using very simple rules like choosing the class corresponding to the largest feature value. We call this classifier the arg max classifier. It is unlikely that this approach is optimal, and in this study, we propose a classification method specifically designed to optimise the performance measure of BCIs. This approach gives an advantage over standard machine learning methods, which aim to optimise different measures. The performance of the proposed algorithm is tested on two publicly available datasets in offline experiments. We use the standard power spectral density analysis (PSDA) and canonical correlation analysis (CCA) feature extraction methods on one dataset and show that the current approach outperforms most of the related studies on this dataset. On the second dataset, we use the task-related component analysis (TRCA) method and demonstrate that the proposed method outperforms the standard argmax classification rule in terms of information transfer rate when using a small number of classes. To our knowledge, this is the first time the information bottleneck is used in the context of SSVEP-based BCIs. The approach is unique in the sense that optimisation is done over the space of classification functions. It potentially improves the performance of BCIs and makes it easier to calibrate the system for different subjects.

POLYNOMIALS WITH ZEROS ON THE UNIT CIRCLE: REGULARITY OF LEJA SEQUENCES

Let z 1 , ⋯ , z m be m distinct complex numbers, normalized to | z k | = 1 , and consider the polynomial p m ( z ) = ∏ k = 1 m ( z − z k ) . We define a sequence of polynomials in a greedy fashion, p N + 1 ( z ) = p N ( z ) z − z ∗ where z ∗ = arg max | z | = 1 | p N ( z ) | , and prove that, independently of the initial polynomial p m , the roots of p N equidistribute in angle at rate at most ( log N ) 2 / N . This persists when sometimes adding “adversarial” points by hand. We also obtain sharp rates for an L2-version of a problem first raised by Erdős and solved by Beck in L ∞ .

Estimation of a distribution function with increasing failure rate average

A life distribution function F is said to have an increasing failure rate average if H(x)∕x is nondecreasing where H(x) is the corresponding cumulative hazard function. In this paper we provide a uniformly strongly consistent estimator of F and derive the convergence in distribution of the estimator at a point where H(x)∕x is increasing using the arg max theorem. We also show using simulations that, unlike other estimators of the past, this new estimator outperforms the empirical distribution in terms of mean square error at all quantiles. An example is also discussed to illustrate the theoretical results

Statistical properties of cerebral CT perfusion imaging systems. Part II. Deconvolution-based systems.

The purpose of this work was to develop a theoretical framework to pinpoint the quantitative relationship between input parameters of deconvolution-based cerebral computed tomography perfusion (CTP) imaging systems and statistical properties of the output perfusion maps. Deconvolution-based CTP systems assume that the arterial input function, tissue enhancement curve, and flow-scaled residue function k(t) are related to each other through a convolution model, and thus by reversing the convolution operation, k(t) and the associated perfusion parameters can be estimated. The theoretical analysis started by deriving analytical formulas for the expected value and autocovariance of the residue function estimated using the singular value decomposition-based deconvolution method. Next, it analyzed statistical properties of the "max" and "arg max" operators, based on which the signal and noise properties of cerebral blood flow (CBF) and time-to-max ( ) are quantitatively related to the statistical model of the estimated residue function [ ] and system parameters. To validate the theory, CTP images of a digital head phantom were simulated, from which signal and noise of each perfusion parameter were measured and compared with values calculated using the theoretical model. In addition, an in vivo canine experiment was performed to validate the noise model of cerebral blood volume (CBV). For the numerical study, the relative root mean squared error between the measured and theoretically calculated value is ≤0.21% for the autocovariance matrix of , and is ≤0.13% for the expected form of . A Bland-Altman analysis demonstrated no significant difference between measured and theoretical values for the mean or noise of each perfusion parameter. For the animal study, the theoretical CBV noise fell within the 25th and 75th percentiles of the experimental values. To provide an example of the theory's utility, an expansion of the CBV noise formula was performed to unveil the dominant role of the baseline image noise in deconvolution-based CBV. Correspondingly, data of the three canine subjects used in the Part I paper were retrospectively processed to confirm that preferentially partitioning dose to the baseline frames benefits both nondeconvolution- and deconvolution-based CBV maps. Quantitative relationships between the statistical properties of deconvolution-based CTP maps, source image acquisition and reconstruction parameters, contrast injection protocol, and deconvolution parameters are established.

Smoothed arg max Extreme Learning Machine: An Alternative to Avoid Classification Ripple in sEMG Signals.

Despite all the recent developments of using the surface electromyography (sEMG) as a control signal, reliable classifications still remain an arduous task due to overlapping classes and classification ripples. In this paper, we present a straightforward approach to avoid classification ripple based on smoothing the arg max value of an Extreme Learning Machine (ELM) classifier. We compare the baseline accuracy of the classifier with an arg max filtered by a traditional Exponential Smoothing Filter (ESF) and our adaptation of Antonyan Vardan Transform (AVT). The classifiers were evaluated using sEMG data acquired through 12 channels from four subjects performing 17 different movements of forearm and fingers with three repetitions each. In the best scenario, our methods reached results higher than 96% and 82% of overall and weighted accuracy, respectively. Those results match or outperform similar papers of the literature using a simpler model, which may help the application of the techniques on embedded platforms and make the practical use of such devices more feasible.

Large-Scale Location-Aware Services in Access: Hierarchical Building/Floor Classification and Location Estimation Using Wi-Fi Fingerprinting Based on Deep Neural Networks

ABSTRACTWe report the results of our investigation on the use of deep neural networks (DNNs) for building/floor classification and floor-level location estimation based on Wi-Fi fingerprinting. We propose a new DNN architecture based on a stacked autoencoder for feature space dimension reduction and a feed-forward classifier for multi-label classification with arg max functions to convert multi-label classification results into multi-class classification ones. We also demonstrate a prototype system for floor-level location estimation using received signal strengths measured on XJTLU campus. Our results show the strengths of DNN-based approaches, providing near state-of-the-art performance with less parameter tuning and higher scalability.

Estimation of a star-shaped distribution function

ABSTRACTA life distribution function (DF) F is said to be star-shaped if is nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point where is increasing using the arg max theorem.

Overhead Performance Tradeoffs—A Resource Allocation Perspective

A key aspect of many resource allocation problems is the need for the resource controller to compute a function, such as the max or arg max, of the competing users metrics. Information must be exchanged between the competing users and the resource controller in order for this function to be computed. In many practical resource controllers the competing users' metrics are communicated to the resource controller, which then computes the desired extremization function. However, in this paper it is shown that information rate savings can be obtained by recognizing that controller only needs to determine the result of this extremization function. If the extremization function is to be computed losslessly, the rate savings are shown in most cases to be at most 2 bits independent of the number of competing users. Motivated by the small savings in the lossless case, simple achievable schemes for both the lossy and interactive variants of this problem are considered. It is shown that both of these approaches have the potential to realize large rate savings, especially in the case where the number of competing users is large. For the lossy variant, it is shown that the proposed simple achievable schemes are in fact close to the fundamental limit given by the rate distortion function.

Erratum to: Testing structural changes in panel data with small fixed panel size and bootstrap

In the original publication, the form of the change point estimate presented in Section 4 is not suitable. The correct form of the estimate with correct assertion of Theorem 3 is given below. The corrected version of the original paper can be found on arXiv http://arxiv.org/abs/1509.01291. Having a sequence of weights {w(t)}T t=2, let us define the estimate of τ as τN := arg min t=2,...,T 1 w(t) N ∑

Orthogonal Matching Pursuit on Faulty Circuits

With the wide recognition that modern nanoscale devices will be error-prone, characterization of reliability of information processing systems built out of unreliable components has become an important topic. In this paper, we analyze the performance of orthogonal matching pursuit (OMP), a popular sparse recovery algorithm, running on faulty circuits. We identify sufficient conditions for correct recovery of the signal support and express these conditions in terms of the relationship among signal magnitudes, sparsity, and the mutual incoherence of the measurement matrix. We study both the effects of additive errors in arithmetic computations and logical errors in comparators. We find that the additive errors in the OMP computations have an impact on the overall performance comparable to that of the additive noise in the input measurements. We also show that parallel structures are more robust to logical errors than serial structures in the implementation of a noisy arg max operation, and thus lead to a better OMP performance.

Analysis of regularized inversion of data corrupted by white Gaussian noise

Tikhonov regularization is studied in the case of linear pseudodifferential operator as the forward map and additive white Gaussian noise as the measurement error. The measurement model for an unknown function u(x) ism(x)=Au(x)+δϵ(x),?>where δ > 0 is the noise magnitude. If ε was an L2-function, Tikhonov regularization gives an estimateTα(m)= arg min u∈Hr∥Au−m∥L22+α∥u∥Hr2?>for u where α = α(δ) is the regularization parameter. Here penalization of the Sobolev norm covers the cases of standard Tikhonov regularization (r = 0) and first derivative penalty (r = 1). Realizations of white Gaussian noise are almost never in L2, but do belong to Hs with probability one if s < 0 is small enough. A modification of Tikhonov regularization theory is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of regularized reconstructions to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. The convergence of the related finite-dimensional problems to the infinite-dimensional problem is also analysed.

On the asymptotic robustness of the likelihood ratio test in quantitative trait locus detection

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (i.e. a gene with quantitative effect on a trait) on a chromosome. We consider two different recombination models. We prove that even if the LRT is constructed from the false recombination model (i.e. the model which does not correspond to the one of the data), the maximum of the LRT process converges asymptotically to the maximum of the LRT process constructed from the true recombination model. We also prove that under some conditions, the arg max of the LRT processes will be different.

Estimation and testing of crossing‐points in fixed design regression

Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing‐point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing‐point, or vice versa. We propose a non‐parametric estimator and show weak and strong consistency as long as the crossing‐point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H0 that the regression function m actually is constant (no crossing‐point), a decision rule is designed for the specific alternative H1 that m possesses a crossing‐point. The pertaining test‐statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov–Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.

Bahadur representation for the nonparametric M-estimator under α-mixing dependence

Under the condition that the observations, which come from a high-dimensional population (X, Y), are strongly stationary and strongly mixing, through using the local linear method, we investigate in this paper, the strong Bahadur representation of the nonparametric M-estimator for the unknown function m(x)=arg min a 𝔼(ρ (a, Y)|X=x), where the loss function ρ(a, y) is measurable. Furthermore, some related simulations are illustrated by using the cross-validation method for both bivariate linear and bivariate nonlinear time series contaminated by heavy-tailed errors. The M-estimator is applied to a series of S&P 500 index futures and spot prices to compare its performance in practice with the ‘usual’ squared-loss regression estimator.

Statistically grounded logic operators in fuzzy sets

Abstract In this study, we are concerned with a category of logic operators for fuzzy sets that is inherently associated with underlying statistical properties of membership grades they operate on. These constructs are referred to as statistically grounded logic operators, namely statistically grounded ORs (SORs, for short) and statistically grounded ANDs (SANDs, for short) operators. In essence, they arise as a solution to the optimization problem of the following form: arg min m ∈ [ 0 , 1 ] ∑ k = 1 N w ( x k ) | x k - m | , where {xk}, k = 1, 2, …, N are the corresponding membership grades to be combined. The weight function w: [0, 1] → [0, 1] captures the nature of the logic and and or aggregation, respectively. We show that the weight function could be induced by t-norms and t-conorms. The weight functions could be also implied by some statistical characteristics of data (membership grades). The choice of the t-norm (t-conorm) depends upon the predefined form of the logic operators to be developed. We demonstrate that SANDs and SORs offer an efficient operational framework for constructing fuzzy rough sets; lead to the increased sensitivity of computing possibility and necessity measures, bring a new insight into fuzzy relational equations and deliver an interpretation vehicle for fuzzy clustering (that is provided in the form of dependency analysis).

Inference for identifiable parameters in partially identified econometric models

This paper considers the problem of inference for partially identified econometric models. The class of models studied are defined by a population objective function Q ( θ , P ) for θ ∈ Θ . The second argument indicates the dependence of the objective function on P , the distribution of the observed data. Unlike the classical extremum estimation framework, it is not assumed that Q ( θ , P ) has a unique minimizer in the parameter space Θ . The goal may be either to draw inferences about some unknown point in the set of minimizers of the population objective function or to draw inferences about the set of minimizers itself. In this paper, the object of interest is some unknown point θ ∈ Θ 0 ( P ) , where Θ 0 ( P ) = arg min θ ∈ Θ Q ( θ , P ) , and so we seek random sets that contain each θ ∈ Θ 0 ( P ) with at least some prespecified probability asymptotically. We also consider situations where the object of interest is the image of some point θ ∈ Θ 0 ( P ) under a known function. Computationally intensive, yet feasible procedures for constructing random sets satisfying the desired coverage property under weak assumptions are provided. We also provide conditions under which the confidence regions are uniformly consistent in level.

Bahadur Representation for the Nonparametric M-Estimator Under alpha-mixing Dependence

Under the condition that the observations, which come from a high-dimensional population (X,Y), are strongly stationary and strongly-mixing, through using the local linear method, we investigate, in this paper, the strong Bahadur representation of the nonparametric M-estimator for the unknown function m(x) = arg min a E(p (a,Y) | X = x), where the loss function p (a,y) is measurable. Furthermore, some related simulations are illustrated by using the cross validation method for both bivariate linear and bivariate nonlinear time series contaminated by heavy-tailed errors. The M-estimator is applied to a series of S&P 500 index futures and spot prices to compare its performance in practice with the usual squared-loss regression estimator.

Asymptotic estimation theory of multipoint linkage analysis under perfect marker information

We consider estimation of a disease susceptibility locus $\tau$ at a chromosome. With perfect marker data available, the estimator $\htau_N$ of $\tau$ based on $N$-pedigrees has a rate of convergence $N^{-1}$ under mild regularity conditions. The limiting distribution is the arg max of a certain compound Poisson process. Our approach is conditional on observed phenotypes, and therefore treats parametric and nonparametric linkage, as well as quantitative trait loci methods within a unified framework. A constant appearing in the asymptotics, the so-called asymptotic slope-to-noise ratio, is introduced as a performance measure for a given genetic model, score function and weighting scheme. This enables us to define asymptotically optimal score functions and weighting schemes. Interestingly, traditional $N^{-1/2}$ theory breaks down, in that, for instance, the ML-estimator is not asymptotically optimal. Further, the asymptotic estimation theory automatically takes uncertainty of $\tau$ into account, which is otherwise handled by means of multiple testing and Bonferroni-type corrections. Other potential applications of our approach that we discuss are general sampling criteria for planning of linkage studies, appropriate grid size of marker maps, robustness w.r.t. choice of map function (dropping assumption of no interference) and quantification of information loss due to heterogeneity (with linked or unlinked trait loci). We also discuss relations to pointwise performance criteria and pay special attention to weak genetic models, so-called local specificity models.

Thurstonian-type representations for “same-different” discriminations: Deterministic decisions and independent images

A discrimination probability function ψ( x , y ) obtained in the “same-different” paradigm assigns to every ordered pair of stimuli ( x , y ) the probability with which they are judged to be different. This function is said to possess the regular minimality property if, for any stimulus pair ( a , b ) , arg min y ψ( a , y )= b ⇔ arg min x ψ( x , b )= a . That is, b is the point of subjective equality for a if and only if a is the point of subjective equality for b . If the value of ψ( a , b ) across all such pairs ( a , b ) is not constant, the function is said to possess the nonconstant self-similarity property. A Thurstonian-type representation for ψ( x , y ) (with independent images and deterministic decisions) is a model in which the two stimuli are mapped into two independent random variables P( x ) and Q( y ) taking on their values in some “perceptual” space; and the decision whether the two stimuli are different is determined by the realizations of the two random variables in a given trial. Thurstonian-type representations can also be called “random utility” ones, provided one imposes no a priori restrictions on the structure of the perceptual space, the distributions of P( x ) and Q( y ) , or the decision rules used. It is shown that (A) any ψ( x , y ) has a Thurstonian-type representation; but (B) if ψ( x , y ) possesses the regular minimality and nonconstant self-similarity properties, it cannot have a “well-behaved” Thurstonian-type representation, in which the probability with which P( x ) , or Q( y ) , falls within a given subset of the perceptual space has appropriately defined bounded directional derivatives with respect to x (respectively, y ). This regularity feature is likely to be found in most conceivable Thurstonian-type models constructed to fit empirical data.

Stabilization Method for Unstable Equilibrium Programming Problems with Coupled Constraints

This point v∗ is called an equilibrium point for the problem (1), (2). The set of equilibrium points for this problem is denoted by V∗. The equilibrium point for problem (1), (2) can be defined in a somewhat different way. Namely, we introduce a multivalued mapping that to each point v ∈ W0, puts in correspondence the set Arg min w∈V (v) Φ(v,w) of minimum points of the function Φ(v,w) with respect to the variable w on the