Researchers have used from 30 days to several years of daily returns as source data for clusterin... more Researchers have used from 30 days to several years of daily returns as source data for clustering financial time series based on their correlations. This paper sets up a statistical framework to study the validity of such practices. We first show that clustering correlated random variables from their observed values is statistically consistent. Then, we also give a first empirical answer to the much debated question: How long should the time series be? If too short, the clusters found can be spurious; if too long, dynamics can be smoothed out.
The following working document summarizes our work on the clustering of financial time series. It... more The following working document summarizes our work on the clustering of financial time series. It was written for a workshop on information geometry and its application for image and signal processing. This workshop brought several experts in pure and applied mathematics together with applied researchers from medical imaging, radar signal processing and finance. The authors belong to the latter group. This document was written as a long introduction to further development of geometric tools in financial applications such as risk or portfolio analysis. Indeed, risk and portfolio analysis essentially rely on covariance matrices. Besides that the Gaussian assumption is known to be inaccurate , covariance matrices are difficult to estimate from empirical data. To filter noise from the empirical estimate, Mantegna proposed using hierarchical clustering. In this work, we first show that this procedure is statistically consistent. Then, we propose to use clustering with a much broader application than the filtering of empirical covariance matrices from the estimate correlation coefficients. To be able to do that, we need to obtain distances between the financial time series that incorporate all the available information in these cross-dependent random processes.
Recently, a large amount of work has been devoted to the study of spectral clustering-a powerful ... more Recently, a large amount of work has been devoted to the study of spectral clustering-a powerful unsupervised classification method. This paper brings contributions to both its foundations, and its applications to text classification. Departing from the mainstream, concerned with hard membership, we study the extension of spectral clustering to soft membership (probabilistic, EM style) assignments. One of its key features is to avoid the complexity gap of hard membership. We apply this theory to a challenging problem, text clustering for languages having permeable borders, via a novel construction of Markov chains from corpora. Experiments with a readily available code clearly display the potential of the method, which brings a visually appealing soft distinction of languages that may define altogether a whole corpus.
Abstract: HyperMask is a system which projects an animated face onto a physical mask, worn by an ... more Abstract: HyperMask is a system which projects an animated face onto a physical mask, worn by an actor. As the mask moves within a prescribed area, its position and orientation are detected by a camera, and the projected image changes with respect to the viewpoint of the audience. The lips of the projected face are automatically synthesized in real time
We propose a meta user interface to design the user interface to tune vector-valued parameters of... more We propose a meta user interface to design the user interface to tune vector-valued parameters of a given parametrized model. In our framework, the user defines a desired number of slider bars to control the parameters. The number of slider bars is independent from the dimension of the original parametrized model, which is important to design perceptually plausible user interface. Those slider bars correspond to straight axes in the parameter space. If the axes are not perpendicular each other, change of the value of one slider bar affects other slider bars, which significantly degrades the usability of the system. Our system can minimize this effect by simply enabling the lock attribute of each slider bar, which orthogonalizes the corresponding axes in the parameter domain. We show an image color adjustment software as our application example. However, the underlying idea of our system can be applied to any parametrized models.
We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagr... more We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power diagram, requiring however algebraic arithmetic. By considering the lesser-known Beltrami hemisphere model of hyperbolic geometry, we overcome the arithmetic limitations of Klein construction. Finally, we characterize the bisectors and geodesics in the other Poincar\' e upper half-space, the Poincar\'e ball, and the Lorentz hyperboloid models, and discusses on degenerate cases for which the dual hyperbolic Delaunay complex is not a triangulation.
In this paper, we present a recommendation system for automatically recentering and cropping digi... more In this paper, we present a recommendation system for automatically recentering and cropping digital still pictures that exhibit capturing artefacts. Autoframing images not only yields better visual pictures but more importantly allows us to remove undesirable artefacts such as lens obstructions by fingers, cellphone straps, or back heads. We report on our real-time prototype system that is targeted to consumer digital still cameras.
Finding mean of matrices becomes increasingly important in modern signal processing problems that... more Finding mean of matrices becomes increasingly important in modern signal processing problems that involve matrix-valued images. In this paper, we define the mean for a set of symmetric positive definite (SPD) matrices based on information-theoretic divergences as the unique minimizer of the averaged divergences, and compare it with the means computed using the Rieman-nian and Log-Euclidean metrics. For the class of divergences induced by the convexity gap of a matrix functional, we present a fast iterative concave-convex optimization scheme with guaranteed convergence to efficiently approximate those divergence-based means.
This paper proposes to abandon one of the most restrictive assumptions often associated with the ... more This paper proposes to abandon one of the most restrictive assumptions often associated with the mean-variance model (namely the Gaussian distribution of stock returns) and to replace it with the exponential families of distributions with matrix parameters. Variance as a criterion of risk assessing is then replaced by divergence modeled as a Bregman matrix. This technique comes from machine learning and quantum physics. The ultimate goal is that of devising algorithms to track efficient portfolios. The authors are able to characterize in the framework of their model some familiar concepts such as risk premium and the certainty equivalent of the expected utility (Theorem 1). The final part of the paper is devoted to learning, i.e., algorithms useful to track reference portfolios, the counterpart in this setting of the familiar efficient portfolios.
More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening conta... more More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening contact between economic production functions (EPFs) -- a cornerstone of neoclassical economics -- and information theory, showing how a generalization of the Cobb-Douglas production function encodes homogeneous functions. As expected by Charnes \textit{et al.}, the contact turns out to be much broader: we show how information geometry as pioneered by Amari and others underpins static and dynamic descriptions of microeconomic cornerstones. We show that the most popular EPFs are fundamentally grounded in a very weak axiomatization of economic transition costs between inputs. The strength of this characterization is surprising, as it geometrically bonds altogether a wealth of collateral economic notions -- advocating for applications in various economic fields --: among all, it characterizes (i) Marshallian and Hicksian demands and their geometric duality, (ii) Slutsky-type properties for the t...
Proceedings of the 2005 symposium on Interactive 3D graphics and games - SI3D '05, 2005
: (a) 105 × 73 × 78 head MRI data. (b) Drawing a 2D stroke along the contour of the brain. (c) Re... more : (a) 105 × 73 × 78 head MRI data. (b) Drawing a 2D stroke along the contour of the brain. (c) Resulting 3D region. The system automatically computes the depth of the stroke and applies constrained segmentation.
We present an approach to multiscale image analysis. It hinges on an operative definition of text... more We present an approach to multiscale image analysis. It hinges on an operative definition of texture that involves a "small region", where some (unknown) statistic is aggregated, and a "large region" within which it is stationary. At each point, multiple small and large regions co-exist at multiple scales, as image structures are pooled by the scaling and quantization process to form "textures" and then transitions between textures define again "structures." We present a technique to learn and agglomerate sparse bases at multiple scales. To do so efficiently, we propose an analysis of cluster statistics after a clustering step is performed, and a new clustering method with linear-time performance. In both cases, we can infer all the "small" and "large" regions at multiple scale in one shot.
We introduce a novel algorithm to learn mixtures of Gamma distributions. This is an extension of ... more We introduce a novel algorithm to learn mixtures of Gamma distributions. This is an extension of the k-Maximum Likelihood estimator algorithm for mixtures of exponential families. Although Gamma distributions are exponential families, we cannot rely directly on the exponential families tools due to the lack of closed-form formula and the cost of numerical approximation: our method uses Gamma distributions with a fixed rate parameter and a special step to choose this parameter is added in the algorithm. Since it converges locally and is computationally faster than an Expectation-Maximization method for Gamma mixture models, our method can be used beneficially as a drop-in replacement in any application using this kind of statistical models.
2010 20th International Conference on Pattern Recognition, 2010
Bhattacharrya distance (BD) is a widely used distance in statistics to compare probability densit... more Bhattacharrya distance (BD) is a widely used distance in statistics to compare probability density functions (PDFs). It has shown strong statistical properties (in terms of Bayes error) and it relates to Fisher information. It has also practical advantages, since it strongly relates on measuring the overlap of the supports of the PDFs. Unfortunately, even with common parametric models on PDFs, few closed-form formulas are known. Moreover, the BD centroid estimation was limited to univariate gaussian PDFs in the literature and no convergence guarantees were provided. In this paper, we propose a closed-form formula for BD on a general class of parametric distributions named exponential families. We show that the BD is a Burbea-Rao divergence for the log normalizer of the exponential family. We propose an efficient iterative scheme to compute a BD centroid on exponential families. Finally, these results allow us to define a Bhattacharrya hierarchical clustering algorithms (BHC). It can be viewed as a generalization of k-means on BD. Results on image segmentation shows the stability of the method.
2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011
The scope of the well-known k-means algorithm has been broadly extended with some recent results:... more The scope of the well-known k-means algorithm has been broadly extended with some recent results: first, the k-means++ initialization method gives some approximation guarantees; second, the Bregman k-means algorithm generalizes the classical algorithm to the large family of Bregman divergences. The Bregman seeding framework combines approximation guarantees with Bregman divergences. We present here an extension of the k-means algorithm using the family of α-divergences. With the framework for representational Bregman divergences, we show that an α-divergence based k-means algorithm can be designed. We present preliminary experiments for clustering and image segmentation applications. Since α-divergences are the natural divergences for constant curvature spaces, these experiments are expected to give information on the structure of the data.
2009 IEEE International Conference on Multimedia and Expo, 2009
Abstract Nearest neighbor (NN) retrieval is a crucial tool of many computer vision tasks. Since t... more Abstract Nearest neighbor (NN) retrieval is a crucial tool of many computer vision tasks. Since the brute-force naive search is too time consuming for most applications, several tailored data structures have been proposed to improve the efficiency of NN search. ...
... In addition, the user can cut the model temporarily and apply modeling operations to the expo... more ... In addition, the user can cut the model temporarily and apply modeling operations to the exposedface to design ... The heart of our technique is automatic guessing of 3D geometry from 2D gestural input, and it is done by making certain assumptions about the target geometry. ...
Researchers have used from 30 days to several years of daily returns as source data for clusterin... more Researchers have used from 30 days to several years of daily returns as source data for clustering financial time series based on their correlations. This paper sets up a statistical framework to study the validity of such practices. We first show that clustering correlated random variables from their observed values is statistically consistent. Then, we also give a first empirical answer to the much debated question: How long should the time series be? If too short, the clusters found can be spurious; if too long, dynamics can be smoothed out.
The following working document summarizes our work on the clustering of financial time series. It... more The following working document summarizes our work on the clustering of financial time series. It was written for a workshop on information geometry and its application for image and signal processing. This workshop brought several experts in pure and applied mathematics together with applied researchers from medical imaging, radar signal processing and finance. The authors belong to the latter group. This document was written as a long introduction to further development of geometric tools in financial applications such as risk or portfolio analysis. Indeed, risk and portfolio analysis essentially rely on covariance matrices. Besides that the Gaussian assumption is known to be inaccurate , covariance matrices are difficult to estimate from empirical data. To filter noise from the empirical estimate, Mantegna proposed using hierarchical clustering. In this work, we first show that this procedure is statistically consistent. Then, we propose to use clustering with a much broader application than the filtering of empirical covariance matrices from the estimate correlation coefficients. To be able to do that, we need to obtain distances between the financial time series that incorporate all the available information in these cross-dependent random processes.
Recently, a large amount of work has been devoted to the study of spectral clustering-a powerful ... more Recently, a large amount of work has been devoted to the study of spectral clustering-a powerful unsupervised classification method. This paper brings contributions to both its foundations, and its applications to text classification. Departing from the mainstream, concerned with hard membership, we study the extension of spectral clustering to soft membership (probabilistic, EM style) assignments. One of its key features is to avoid the complexity gap of hard membership. We apply this theory to a challenging problem, text clustering for languages having permeable borders, via a novel construction of Markov chains from corpora. Experiments with a readily available code clearly display the potential of the method, which brings a visually appealing soft distinction of languages that may define altogether a whole corpus.
Abstract: HyperMask is a system which projects an animated face onto a physical mask, worn by an ... more Abstract: HyperMask is a system which projects an animated face onto a physical mask, worn by an actor. As the mask moves within a prescribed area, its position and orientation are detected by a camera, and the projected image changes with respect to the viewpoint of the audience. The lips of the projected face are automatically synthesized in real time
We propose a meta user interface to design the user interface to tune vector-valued parameters of... more We propose a meta user interface to design the user interface to tune vector-valued parameters of a given parametrized model. In our framework, the user defines a desired number of slider bars to control the parameters. The number of slider bars is independent from the dimension of the original parametrized model, which is important to design perceptually plausible user interface. Those slider bars correspond to straight axes in the parameter space. If the axes are not perpendicular each other, change of the value of one slider bar affects other slider bars, which significantly degrades the usability of the system. Our system can minimize this effect by simply enabling the lock attribute of each slider bar, which orthogonalizes the corresponding axes in the parameter domain. We show an image color adjustment software as our application example. However, the underlying idea of our system can be applied to any parametrized models.
We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagr... more We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power diagram, requiring however algebraic arithmetic. By considering the lesser-known Beltrami hemisphere model of hyperbolic geometry, we overcome the arithmetic limitations of Klein construction. Finally, we characterize the bisectors and geodesics in the other Poincar\' e upper half-space, the Poincar\'e ball, and the Lorentz hyperboloid models, and discusses on degenerate cases for which the dual hyperbolic Delaunay complex is not a triangulation.
In this paper, we present a recommendation system for automatically recentering and cropping digi... more In this paper, we present a recommendation system for automatically recentering and cropping digital still pictures that exhibit capturing artefacts. Autoframing images not only yields better visual pictures but more importantly allows us to remove undesirable artefacts such as lens obstructions by fingers, cellphone straps, or back heads. We report on our real-time prototype system that is targeted to consumer digital still cameras.
Finding mean of matrices becomes increasingly important in modern signal processing problems that... more Finding mean of matrices becomes increasingly important in modern signal processing problems that involve matrix-valued images. In this paper, we define the mean for a set of symmetric positive definite (SPD) matrices based on information-theoretic divergences as the unique minimizer of the averaged divergences, and compare it with the means computed using the Rieman-nian and Log-Euclidean metrics. For the class of divergences induced by the convexity gap of a matrix functional, we present a fast iterative concave-convex optimization scheme with guaranteed convergence to efficiently approximate those divergence-based means.
This paper proposes to abandon one of the most restrictive assumptions often associated with the ... more This paper proposes to abandon one of the most restrictive assumptions often associated with the mean-variance model (namely the Gaussian distribution of stock returns) and to replace it with the exponential families of distributions with matrix parameters. Variance as a criterion of risk assessing is then replaced by divergence modeled as a Bregman matrix. This technique comes from machine learning and quantum physics. The ultimate goal is that of devising algorithms to track efficient portfolios. The authors are able to characterize in the framework of their model some familiar concepts such as risk premium and the certainty equivalent of the expected utility (Theorem 1). The final part of the paper is devoted to learning, i.e., algorithms useful to track reference portfolios, the counterpart in this setting of the familiar efficient portfolios.
More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening conta... more More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening contact between economic production functions (EPFs) -- a cornerstone of neoclassical economics -- and information theory, showing how a generalization of the Cobb-Douglas production function encodes homogeneous functions. As expected by Charnes \textit{et al.}, the contact turns out to be much broader: we show how information geometry as pioneered by Amari and others underpins static and dynamic descriptions of microeconomic cornerstones. We show that the most popular EPFs are fundamentally grounded in a very weak axiomatization of economic transition costs between inputs. The strength of this characterization is surprising, as it geometrically bonds altogether a wealth of collateral economic notions -- advocating for applications in various economic fields --: among all, it characterizes (i) Marshallian and Hicksian demands and their geometric duality, (ii) Slutsky-type properties for the t...
Proceedings of the 2005 symposium on Interactive 3D graphics and games - SI3D '05, 2005
: (a) 105 × 73 × 78 head MRI data. (b) Drawing a 2D stroke along the contour of the brain. (c) Re... more : (a) 105 × 73 × 78 head MRI data. (b) Drawing a 2D stroke along the contour of the brain. (c) Resulting 3D region. The system automatically computes the depth of the stroke and applies constrained segmentation.
We present an approach to multiscale image analysis. It hinges on an operative definition of text... more We present an approach to multiscale image analysis. It hinges on an operative definition of texture that involves a "small region", where some (unknown) statistic is aggregated, and a "large region" within which it is stationary. At each point, multiple small and large regions co-exist at multiple scales, as image structures are pooled by the scaling and quantization process to form "textures" and then transitions between textures define again "structures." We present a technique to learn and agglomerate sparse bases at multiple scales. To do so efficiently, we propose an analysis of cluster statistics after a clustering step is performed, and a new clustering method with linear-time performance. In both cases, we can infer all the "small" and "large" regions at multiple scale in one shot.
We introduce a novel algorithm to learn mixtures of Gamma distributions. This is an extension of ... more We introduce a novel algorithm to learn mixtures of Gamma distributions. This is an extension of the k-Maximum Likelihood estimator algorithm for mixtures of exponential families. Although Gamma distributions are exponential families, we cannot rely directly on the exponential families tools due to the lack of closed-form formula and the cost of numerical approximation: our method uses Gamma distributions with a fixed rate parameter and a special step to choose this parameter is added in the algorithm. Since it converges locally and is computationally faster than an Expectation-Maximization method for Gamma mixture models, our method can be used beneficially as a drop-in replacement in any application using this kind of statistical models.
2010 20th International Conference on Pattern Recognition, 2010
Bhattacharrya distance (BD) is a widely used distance in statistics to compare probability densit... more Bhattacharrya distance (BD) is a widely used distance in statistics to compare probability density functions (PDFs). It has shown strong statistical properties (in terms of Bayes error) and it relates to Fisher information. It has also practical advantages, since it strongly relates on measuring the overlap of the supports of the PDFs. Unfortunately, even with common parametric models on PDFs, few closed-form formulas are known. Moreover, the BD centroid estimation was limited to univariate gaussian PDFs in the literature and no convergence guarantees were provided. In this paper, we propose a closed-form formula for BD on a general class of parametric distributions named exponential families. We show that the BD is a Burbea-Rao divergence for the log normalizer of the exponential family. We propose an efficient iterative scheme to compute a BD centroid on exponential families. Finally, these results allow us to define a Bhattacharrya hierarchical clustering algorithms (BHC). It can be viewed as a generalization of k-means on BD. Results on image segmentation shows the stability of the method.
2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011
The scope of the well-known k-means algorithm has been broadly extended with some recent results:... more The scope of the well-known k-means algorithm has been broadly extended with some recent results: first, the k-means++ initialization method gives some approximation guarantees; second, the Bregman k-means algorithm generalizes the classical algorithm to the large family of Bregman divergences. The Bregman seeding framework combines approximation guarantees with Bregman divergences. We present here an extension of the k-means algorithm using the family of α-divergences. With the framework for representational Bregman divergences, we show that an α-divergence based k-means algorithm can be designed. We present preliminary experiments for clustering and image segmentation applications. Since α-divergences are the natural divergences for constant curvature spaces, these experiments are expected to give information on the structure of the data.
2009 IEEE International Conference on Multimedia and Expo, 2009
Abstract Nearest neighbor (NN) retrieval is a crucial tool of many computer vision tasks. Since t... more Abstract Nearest neighbor (NN) retrieval is a crucial tool of many computer vision tasks. Since the brute-force naive search is too time consuming for most applications, several tailored data structures have been proposed to improve the efficiency of NN search. ...
... In addition, the user can cut the model temporarily and apply modeling operations to the expo... more ... In addition, the user can cut the model temporarily and apply modeling operations to the exposedface to design ... The heart of our technique is automatic guessing of 3D geometry from 2D gestural input, and it is done by making certain assumptions about the target geometry. ...
Uploads
Papers by Frank Nielsen