arXiv subjects

Frank Nielsen

Publications and source records attributed to Frank Nielsen.

At least 127 records · Page 7Linked to original sources

arXiv2016

Optimal Copula Transport for Clustering Multivariate Time Series

This paper presents a new methodology for clustering multivariate time series leveraging optimal transport between copulas. Copulas are used to encode both (i) intra-dependence of a multivariate time series, and (ii) inter-dependence between two time series. Then, optimal copula transport allows us to define two distances between multivariate time series: (i) one for measuring intra-dependence dissimilarity, (ii) another one for measuring inter-dependence dissimilarity based on a new multivariate dependence coefficient which is robust to noise, deterministic, and which can target specified dependencies.

cs.LG↗

arXiv2015

A proposal of a methodological framework with experimental guidelines to investigate clustering stability on financial time series

We present in this paper an empirical framework motivated by the practitioner point of view on stability. The goal is to both assess clustering validity and yield market insights by providing through the data perturbations we propose a multi-view of the assets' clustering behaviour. The perturbation framework is illustrated on an extensive credit default swap time series database available online at www.datagrapple.com.

q-fin.ST↗

arXiv2015

Comment partitionner automatiquement des marches aléatoires ? Avec application à la finance quantitative

We present in this paper a novel non-parametric approach useful for clustering Markov processes. We introduce a pre-processing step consisting in mapping multivariate independent and identically distributed samples from random variables to a generic non-parametric representation which factorizes dependency and marginal distribution apart without losing any. An associated metric is defined where the balance between random variables dependency and distribution information is controlled by a single parameter. This mixing parameter can be learned or played with by a practitioner, such use is illustrated on the case of clustering financial time series. Experiments, implementation and results obtained on public financial time series are online on a web portal \url{http://www.datagrapple.com}.

cs.CE↗

arXiv2015

On conformal divergences and their population minimizers

Total Bregman divergences are a recent tweak of ordinary Bregman divergences originally motivated by applications that required invariance by rotations. They have displayed superior results compared to ordinary Bregman divergences on several clustering, computer vision, medical imaging and machine learning tasks. These preliminary results raise two important problems : First, report a complete characterization of the left and right population minimizers for this class of total Bregman divergences. Second, characterize a principled superset of total and ordinary Bregman divergences with good clustering properties, from which one could tailor the choice of a divergence to a particular application. In this paper, we provide and study one such superset with interesting geometric features, that we call conformal divergences, and focus on their left and right population minimizers. Our results are obtained in a recently coined $(u, v)$-geometric structure that is a generalization of the dually flat affine connections in information geometry. We characterize both analytically and geometrically the population minimizers. We prove that conformal divergences (resp. total Bregman divergences) are essentially exhaustive for their left (resp. right) population minimizers. We further report new results and extend previous results on the robustness to outliers of the left and right population minimizers, and discuss the role of the $(u, v)$-geometric structure in clustering. Additional results are also given.

cs.IT↗

arXiv2014

Further results on the hyperbolic Voronoi diagrams

In Euclidean geometry, it is well-known that the $k$-order Voronoi diagram in $\mathbb{R}^d$ can be computed from the vertical projection of the $k$-level of an arrangement of hyperplanes tangent to a convex potential function in $\mathbb{R}^{d+1}$: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a {\em concave} potential function: the northern hemisphere. Furthermore, we also show how to build the hyperbolic $k$-order diagrams as equivalent clipped power diagrams in $\mathbb{R}^d$. We investigate the hyperbolic Voronoi diagram in the hyperboloid model and show how it reduces to a Klein-type model using central projections.

cs.CG↗

arXiv2014

Further heuristics for $k$-means: The merge-and-split heuristic and the $(k,l)$-means

Finding the optimal $k$-means clustering is NP-hard in general and many heuristics have been designed for minimizing monotonically the $k$-means objective. We first show how to extend Lloyd's batched relocation heuristic and Hartigan's single-point relocation heuristic to take into account empty-cluster and single-point cluster events, respectively. Those events tend to increasingly occur when $k$ or $d$ increases, or when performing several restarts. First, we show that those special events are a blessing because they allow to partially re-seed some cluster centers while further minimizing the $k$-means objective function. Second, we describe a novel heuristic, merge-and-split $k$-means, that consists in merging two clusters and splitting this merged cluster again with two new centers provided it improves the $k$-means objective. This novel heuristic can improve Hartigan's $k$-means when it has converged to a local minimum. We show empirically that this merge-and-split $k$-means improves over the Hartigan's heuristic which is the {\em de facto} method of choice. Finally, we propose the $(k,l)$-means objective that generalizes the $k$-means objective by associating the data points to their $l$ closest cluster centers, and show how to either directly convert or iteratively relax the $(k,l)$-means into a $k$-means in order to reach better local minima.

cs.LG↗

arXiv2014

Optimal interval clustering: Application to Bregman clustering and statistical mixture learning

We present a generic dynamic programming method to compute the optimal clustering of $n$ scalar elements into $k$ pairwise disjoint intervals. This case includes 1D Euclidean $k$-means, $k$-medoids, $k$-medians, $k$-centers, etc. We extend the method to incorporate cluster size constraints and show how to choose the appropriate $k$ by model selection. Finally, we illustrate and refine the method on two case studies: Bregman clustering and statistical mixture learning maximizing the complete likelihood.

cs.IT↗

arXiv2014

On the symmetrical Kullback-Leibler Jeffreys centroids

Due to the success of the bag-of-word modeling paradigm, clustering histograms has become an important ingredient of modern information processing. Clustering histograms can be performed using the celebrated $k$-means centroid-based algorithm. From the viewpoint of applications, it is usually required to deal with symmetric distances. In this letter, we consider the Jeffreys divergence that symmetrizes the Kullback-Leibler divergence, and investigate the computation of Jeffreys centroids. We first prove that the Jeffreys centroid can be expressed analytically using the Lambert $W$ function for positive histograms. We then show how to obtain a fast guaranteed approximation when dealing with frequency histograms. Finally, we conclude with some remarks on the $k$-means histogram clustering.

cs.IT↗

arXiv2014

Generalized Bhattacharyya and Chernoff upper bounds on Bayes error using quasi-arithmetic means

Bayesian classification labels observations based on given prior information, namely class-a priori and class-conditional probabilities. Bayes' risk is the minimum expected classification cost that is achieved by the Bayes' test, the optimal decision rule. When no cost incurs for correct classification and unit cost is charged for misclassification, Bayes' test reduces to the maximum a posteriori decision rule, and Bayes risk simplifies to Bayes' error, the probability of error. Since calculating this probability of error is often intractable, several techniques have been devised to bound it with closed-form formula, introducing thereby measures of similarity and divergence between distributions like the Bhattacharyya coefficient and its associated Bhattacharyya distance. The Bhattacharyya upper bound can further be tightened using the Chernoff information that relies on the notion of best error exponent. In this paper, we first express Bayes' risk using the total variation distance on scaled distributions. We then elucidate and extend the Bhattacharyya and the Chernoff upper bound mechanisms using generalized weighted means. We provide as a byproduct novel notions of statistical divergences and affinity coefficients. We illustrate our technique by deriving new upper bounds for the univariate Cauchy and the multivariate $t$-distributions, and show experimentally that those bounds are not too distant to the computationally intractable Bayes' error.

cs.CV↗

arXiv2013

Total Jensen divergences: Definition, Properties and k-Means++ Clustering

We present a novel class of divergences induced by a smooth convex function called total Jensen divergences. Those total Jensen divergences are invariant by construction to rotations, a feature yielding regularization of ordinary Jensen divergences by a conformal factor. We analyze the relationships between this novel class of total Jensen divergences and the recently introduced total Bregman divergences. We then proceed by defining the total Jensen centroids as average distortion minimizers, and study their robustness performance to outliers. Finally, we prove that the k-means++ initialization that bypasses explicit centroid computations is good enough in practice to guarantee probabilistically a constant approximation factor to the optimal k-means clustering.

cs.IT↗

arXiv2013

On the Chi square and higher-order Chi distances for approximating f-divergences

We report closed-form formula for calculating the Chi square and higher-order Chi distances between statistical distributions belonging to the same exponential family with affine natural space, and instantiate those formula for the Poisson and isotropic Gaussian families. We then describe an analytic formula for the $f$-divergences based on Taylor expansions and relying on an extended class of Chi-type distances.

cs.IT↗

arXiv2013

Logging safely in public spaces using color PINs

Nowadays, we are increasingly logging on many different Internet sites to access private data like emails or photos remotely stored in the clouds. This makes us all the more concerned with digital identity theft and passwords being stolen either by key loggers or shoulder-surfing attacks. Quite surprisingly, the current bottleneck of computer security when logging for authentication is the User Interface (UI): How can we enter safely secret passwords when concealed spy cameras or key loggers may be recording the login session? Logging safely requires to design a secure Human Computer Interface (HCI) robust to those attacks. We describe a novel method and system based on entering secret ID passwords by means of associative secret UI passwords that provides zero-knowledge to observers. We demonstrate the principles using a color Personal Identification Numbers (PINs) login system and describes its various extensions.

cs.HC↗

arXiv2013

Cramer-Rao Lower Bound and Information Geometry

This article focuses on an important piece of work of the world renowned Indian statistician, Calyampudi Radhakrishna Rao. In 1945, C. R. Rao (25 years old then) published a pathbreaking paper, which had a profound impact on subsequent statistical research.

cs.IT↗

arXiv2012

The Burbea-Rao and Bhattacharyya centroids

We study the centroid with respect to the class of information-theoretic Burbea-Rao divergences that generalize the celebrated Jensen-Shannon divergence by measuring the non-negative Jensen difference induced by a strictly convex and differentiable function. Although those Burbea-Rao divergences are symmetric by construction, they are not metric since they fail to satisfy the triangle inequality. We first explain how a particular symmetrization of Bregman divergences called Jensen-Bregman distances yields exactly those Burbea-Rao divergences. We then proceed by defining skew Burbea-Rao divergences, and show that skew Burbea-Rao divergences amount in limit cases to compute Bregman divergences. We then prove that Burbea-Rao centroids are unique, and can be arbitrarily finely approximated by a generic iterative concave-convex optimization algorithm with guaranteed convergence property. In the second part of the paper, we consider the Bhattacharyya distance that is commonly used to measure overlapping degree of probability distributions. We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a Burbea-Rao divergence in disguise. Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or "diagonal" multivariate Gaussians. To illustrate the performance of our Bhattacharyya/Burbea-Rao centroid algorithm, we present experimental performance results for $k$-means and hierarchical clustering methods of Gaussian mixture models.

cs.IT↗

arXiv2012

On Approximating the Riemannian 1-Center

In this paper, we generalize the simple Euclidean 1-center approximation algorithm of Badoiu and Clarkson (2003) to Riemannian geometries and study accordingly the convergence rate. We then show how to instantiate this generic algorithm to two particular cases: (1) hyperbolic geometry, and (2) Riemannian manifold of symmetric positive definite matrices.

cs.CG↗

arXiv2012

$k$-MLE: A fast algorithm for learning statistical mixture models

We describe $k$-MLE, a fast and efficient local search algorithm for learning finite statistical mixtures of exponential families such as Gaussian mixture models. Mixture models are traditionally learned using the expectation-maximization (EM) soft clustering technique that monotonically increases the incomplete (expected complete) likelihood. Given prescribed mixture weights, the hard clustering $k$-MLE algorithm iteratively assigns data to the most likely weighted component and update the component models using Maximum Likelihood Estimators (MLEs). Using the duality between exponential families and Bregman divergences, we prove that the local convergence of the complete likelihood of $k$-MLE follows directly from the convergence of a dual additively weighted Bregman hard clustering. The inner loop of $k$-MLE can be implemented using any $k$-means heuristic like the celebrated Lloyd's batched or Hartigan's greedy swap updates. We then show how to update the mixture weights by minimizing a cross-entropy criterion that implies to update weights by taking the relative proportion of cluster points, and reiterate the mixture parameter update and mixture weight update processes until convergence. Hard EM is interpreted as a special case of $k$-MLE when both the component update and the weight update are performed successively in the inner loop. To initialize $k$-MLE, we propose $k$-MLE++, a careful initialization of $k$-MLE guaranteeing probabilistically a global bound on the best possible complete likelihood.

cs.LG↗

arXiv2011

A family of statistical symmetric divergences based on Jensen's inequality

We introduce a novel parametric family of symmetric information-theoretic distances based on Jensen's inequality for a convex functional generator. In particular, this family unifies the celebrated Jeffreys divergence with the Jensen-Shannon divergence when the Shannon entropy generator is chosen. We then design a generic algorithm to compute the unique centroid defined as the minimum average divergence. This yields a smooth family of centroids linking the Jeffreys to the Jensen-Shannon centroid. Finally, we report on our experimental results.

cs.CV↗

arXiv2011

Medians and means in Finsler geometry

We investigate existence and uniqueness of p-means and the median of a probability measure on a Finsler manifold, in relation with the convexity of the support of the measure. We prove that the p-mean is the limit point of a continuous time gradient flow. Under some additional condition which is always satisfied for larger than or equal to 2, a discretization of this path converges to the p-mean. This provides an algorithm for determining those Finsler center points.

math.DG↗