This paper tackles the problem of robust covariance matrix estimation when the data is incomplete. Classical statistical estimation methodologies are usually built upon the Gaussian assumption, whereas existing robust estimation ones assume unstructured signal models. The former can be inaccurate in real-world data sets in which heterogeneity causes heavy-tail distributions, while the latter does not profit from the usual low-rank structure of the signal. Taking advantage of both worlds, a covariance matrix estimation procedure is designed on a robust (compound Gaussian) low-rank model by leveraging the observed-data likelihood function within an expectation-maximization algorithm. It is also designed to handle general pattern of missing values. The proposed procedure is first validated on simulated data sets. Then, its interest for classification and clustering applications is assessed on two real data sets with missing values, which include multispectral and hyperspectral time series.
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Density-based clustering algorithms are widely used for discovering clusters in pattern recognition and machine learning since they can deal with non-hyperspherical clusters and are robustness to handle outliers. However, the runtime of density-based algorithms is heavily dominated by finding neighbors and calculating the density of each point which is time-consuming. To address this issue, this paper proposes a density-based clustering framework by using the fast principal component analysis, which can be applied to density based methods to prune unnecessary distance calculations when finding neighbors and estimating densities. By applying this clustering framework to the Density Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm, an improved DBSCAN (called IDBSCAN) is obtained, which preserves the advantage of DBSCAN and meanwhile, greatly reduces the computation of redundant distances. Experiments on five benchmark datasets demonstrate that the proposed IDBSCAN algorithm improves the computational efficiency significantly.
Extreme-value copulas arise as the limiting dependence structure of component-wise maxima. Defined in terms of a functional parameter, they are one of the most widespread copula families due to its flexibility and ability to capture asymmetry. Despite this, meeting the complex analytical properties of this parameter in an unconstrained setting still remains a challenge, restricting most uses to either models with very few parameters or non-parametric models. On this paper we focus on the bivariate case and propose a novel approach for estimating this functional parameter in a semiparametric manner. Our procedure relies on a series of basic transformations starting from a zero-integral spline. Spline coordinates are fit through maximum likelihood estimation, leveraging gradient optimization, without imposing further constraints. We conduct several experiments on both simulated and real data. Specifically, we test our method on scarce data gathered by the gravitational wave detection LIGO and Virgo collaborations.
Low-rank matrix approximation is one of the central concepts in machine learning, with applications in dimension reduction, de-noising, multivariate statistical methodology, and many more. A recent extension to LRMA is called low-rank matrix completion (LRMC). It solves the LRMA problem when some observations are missing and is especially useful for recommender systems. In this paper, we consider an element-wise weighted generalization of LRMA. The resulting weighted low-rank matrix approximation technique therefore covers LRMC as a special case with binary weights. WLRMA has many applications. For example, it is an essential component of GLM optimization algorithms, where an exponential family is used to model the entries of a matrix, and the matrix of natural parameters admits a low-rank structure. We propose an algorithm for solving the weighted problem, as well as two acceleration techniques. Further, we develop a non-SVD modification of the proposed algorithm that is able to handle extremely high-dimensional data. We compare the performance of all the methods on a small simulation example as well as a real-data application.
Three robust methods for clustering multivariate time series from the point of view of generating processes are proposed. The procedures are robust versions of a fuzzy C-means model based on: (i) estimates of the quantile cross-spectral density and (ii) the classical principal component analysis. Robustness to the presence of outliers is achieved by using the so-called metric, noise and trimmed approaches. The metric approach incorporates in the objective function a distance measure aimed at neutralizing the effect of the outliers, the noise approach builds an artificial cluster expected to contain the outlying series and the trimmed approach eliminates the most atypical series in the dataset. All the proposed techniques inherit the nice properties of the quantile cross-spectral density, as being able to uncover general types of dependence. Results from a broad simulation study including multivariate linear, nonlinear and GARCH processes indicate that the algorithms are substantially effective in coping with the presence of outlying series (i.e., series exhibiting a dependence structure different from that of the majority), clearly poutperforming alternative procedures. The usefulness of the suggested methods is highlighted by means of two specific applications regarding financial and environmental series.
This paper studies distributed binary test of statistical independence under communication (information bits) constraints. While testing independence is very relevant in various applications, distributed independence test is particularly useful for event detection in sensor networks where data correlation often occurs among observations of devices in the presence of a signal of interest. By focusing on the case of two devices because of their tractability, we begin by investigating conditions on Type I error probability restrictions under which the minimum Type II error admits an exponential behavior with the sample size. Then, we study the finite sample-size regime of this problem. We derive new upper and lower bounds for the gap between the minimum Type II error and its exponential approximation under different setups, including restrictions imposed on the vanishing Type I error probability. Our theoretical results shed light on the sample-size regimes at which approximations of the Type II error probability via error exponents became informative enough in the sense of predicting well the actual error probability. We finally discuss an application of our results where the gap is evaluated numerically, and we show that exponential approximations are not only tractable but also a valuable proxy for the Type II probability of error in the finite-length regime.
In a widely studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. For many important multi-parametric optimization problems, an optimal solutions set with minimum cardinality can contain super-polynomially many solutions. Consequently, any exact algorithm for such problems must output a super-polynomial number of solutions. We propose an approximation algorithm that is applicable to a general class of multi-parametric optimization problems and outputs a number of solutions that is bounded polynomially in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric formulations, providing an approximation guarantee that is arbitrarily close to the approximation guarantee for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, the method yields an FPTAS. We discuss implications to important multi-parametric combinatorial optimizations problems. Remarkably, we obtain a $(\frac{3}{2} + \varepsilon)$-approximation algorithm for the multi-parametric metric travelling salesman problem, whereas the non-parametric version is known to be APX-complete. Furthermore, we show that the cardinality of a minimal size approximation set is in general not $\ell$-approximable for any natural number $\ell$.
The probability distribution of precipitation amount strongly depends on geography, climate zone, and time scale considered. Closed-form parametric probability distributions are not sufficiently flexible to provide accurate and universal models for precipitation amount over different time scales. In this paper we derive non-parametric estimates of the cumulative distribution function (CDF) of precipitation amount for wet time intervals. The CDF estimates are obtained by integrating the kernel density estimator leading to semi-explicit CDF expressions for different kernel functions. We investigate kernel-based CDF estimation with an adaptive plug-in bandwidth (KCDE), using both synthetic data sets and reanalysis precipitation data from the island of Crete (Greece). We show that KCDE provides better estimates of the probability distribution than the standard empirical (staircase) estimate and kernel-based estimates that use the normal reference bandwidth. We also demonstrate that KCDE enables the simulation of non-parametric precipitation amount distributions by means of the inverse transform sampling method.
We present a generalization of the Cauchy/Lorentzian, Geman-McClure, Welsch/Leclerc, generalized Charbonnier, Charbonnier/pseudo-Huber/L1-L2, and L2 loss functions. By introducing robustness as a continous parameter, our loss function allows algorithms built around robust loss minimization to be generalized, which improves performance on basic vision tasks such as registration and clustering. Interpreting our loss as the negative log of a univariate density yields a general probability distribution that includes normal and Cauchy distributions as special cases. This probabilistic interpretation enables the training of neural networks in which the robustness of the loss automatically adapts itself during training, which improves performance on learning-based tasks such as generative image synthesis and unsupervised monocular depth estimation, without requiring any manual parameter tuning.
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.
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.
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