Gleb Nosovskiy | Moscow State University (original) (raw)

Papers by Gleb Nosovskiy

Chebyshevskii Sbornik

In the present article we propose a modification of the PaDiM anomaly detection method which maps... more In the present article we propose a modification of the PaDiM anomaly detection method which maps images to vectors and then calculates the Mahalanobis distance between such vectors and the distribution of the vectors of the training set. Of the coordinate axes of the vectors we choose a subset of such that the distribution along them is close to normal according to the chosen statistical criterion. The uniformization procedure is then applied to those coordinates and the Mahalanobis distance is calculated. This approach is shown to increase the ROCAUC value in comparison with the PaDiM method.

Integrability and Nonintegrability in Geometry and Mechanics, 1988

It isn't that they can't see the solution. It is that they can't see the problem. G.K. Chesterton... more It isn't that they can't see the solution. It is that they can't see the problem. G.K. Chesterton. The Scandal of Father Brown 'The point of a Pin' .

2007 International Conference on Cyberworlds (CW'07), 2007

ABSTRACT The mathematical structure of cyberworlds is clarified based on the duality of homology ... more ABSTRACT The mathematical structure of cyberworlds is clarified based on the duality of homology lifting property and homotopy extension property. The duality gives bottom-up and top-down methods to model, design and analyze the structure of cyberworlds. The set of homepages representing a cyberworld is transformed into a state finite machine. In development of the cyberworld, a sequence of finite state machines is obtained. This sequence has homotopic property. This property is clarified to map a finite state machine to a simplicial complex. Wikipedia, bottom-up network construction and top-down network analysis are described as examples.

Error estimate for the direct algorithm of determination of the projective mapping is presented. ... more Error estimate for the direct algorithm of determination of the projective mapping is presented. Such estimates are important in the problem of gluing of 2D projective images obtained from different points in the space [1], [2], [5]. The obtained estimate suggests that the direct algorithm of projective mapping calculation is rather accurate and robust.

Manifold hypothesis states that data points in high-dimensional space actually lie in close vicin... more Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and semi-supervised learning. Here we present new approach to manifold hypothesis checking and underlying manifold dimension estimation. In order to do it we use two very different methods simultaneously - one geometric, another probabilistic - and check whether they give the same result. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. The probabilistic method is new. Although it exploits standard nearest neighborhood distance, it is different from methods which were previously used in such situations. This method is robust, fast and includes special preliminary data transformation. Experiments on real datasets show that the suggested approach based on two methods combination...

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight... more Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier- Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Thus, the papers published in this special issue address topics related to the Navier-Stokes equations, transition to turbulence, and some other related problems. These papers cover topics in applied mathematics, and their applications to science and engineering.

Journal | MESA, 2013

Method of Laplace transform is standard in mathematical physics. One of its advantages is that it... more Method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, Theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.

Manifold hypothesis states that data points in highdimensional space actually lie in close vicini... more Manifold hypothesis states that data points in highdimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and semi-supervised learning. Here we present new approach to manifold hypothesis checking and underlying manifold dimension estimation. In order to do it we use two very different methods simultaneously — one geometric, another probabilistic — and check whether they give the same result. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. The probabilistic method is new. Although it exploits standard nearest neighborhood distance, it is different from methods which were previously used in such situations. This method is robust, fast and includes special preliminary data transformation. Experiments on real datasets show that the suggested approach based on two methods combination ...

Formal analysis and computer recognition of 2D color images is important branch of modern compute... more Formal analysis and computer recognition of 2D color images is important branch of modern computer geometry. However, existing algorithms, although they are highly developed, are not quite satisfactory and seem to be much worse than (unknown) algorithms, which our brain uses to analyze eye information. Almost all existing algorithms omit colors and deal with grayscale transformations only. But in many cases color information is important. In this paper fundamentally new method of coding and analyzing color digital images is suggested. The main point of this method is that a full-color digital image is represented, without dropping colors, by special 2D surface in 3D space, after which it is analyzed by methods of differential geometry, rather than traditional gradient-based or Hessian-based methods (like in SIFT, GLOH, SURF, Canny operator, and many other algorithms).

Pattern Recognition, Sep 1, 2008

Clustering became a classical problem in databases, data warehouses, pattern recognition, artific... more Clustering became a classical problem in databases, data warehouses, pattern recognition, artificial intelligence, and computer graphics. Applications in large spatial databases, point-based graphics, etc., give rise to new requirements for the clustering algorithms: automatic discovering of arbitrary shaped and/or non-homogeneous clusters, discovering of clusters located in low-dimensional hyperspace, detecting cluster boundaries. On that account, a new clustering and boundary detecting algorithm, ADACLUS, is proposed. It is based on the specially constructed adaptive influence function, and therefore, discovers clusters of arbitrary shapes and diverse densities, adequately captures clusters boundaries, and it is robust to noise. Normally ADACLUS performs clustering purely automatically without any preliminary parameter settings. But it also gives the user an optional possibility to set three parameters with clear meaning in order to adjust clustering for special applications. The algorithm was tested on various two-dimensional data sets, and it exhibited its effectiveness in discovering clusters of complex shapes and diverse densities. Linear complexity of the ADACLUS gives it an advantage over some well-known algorithms.

Pattern Recognition Letters, Jul 1, 2008

In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is pro... more In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is proposed. This algorithm treats clustering task by analyzing statistical features of data. For each data point, its values of statistical features are extracted from its neighborhood which effectively models the data proximity. By applying specifically built criteria function, TRICLUST is able to effectively handle data set with clusters of complex shapes and non-uniform densities, and with large amount of noises. One additional advantage of TRICLUST is the boundary detection function which is valuable for many real world applications such as geo-spatial data processing, point-based computer graphics, etc.

A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new c... more A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new concept of the informational channel, viewed as an elementary socio-economic cell. The processes, taking place within this cell, are described by a set of partial differential equations that relate the two main aspects of a socio-economic system: energy (any material asset, such as money, material property, etc.) and information. The solution is obtained in the form of the Volterra integral equation that relates the value of disturbance upon the socio-economical cell and the value of the wealth function, which is defined as an integral characteristic of both material and informational components. Two basic cases of disturbance were analyzed: a constant and a single-pulse flux. In both cases, the results show a quite natural behavior. The proposed model might serve as a basis for more advanced analysis of socio-economical systems.

From the text: Waves follow our boat as we meander across the lake, and turbulent air currents fo... more From the text: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Thus, the papers published in this special issue address topics related to the Navier-Stokes equations, transition to turbulence, and some other related problems. These papers cover topics in applied mathematics, and their applications to science and engineering.

The method of Laplace transform is standard in mathematical physics. One of its advantages is tha... more The method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, a theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.

2007 International Conference on Cyberworlds (CW'07), 2007

Pattern Recognition Letters, 2008

In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is pro... more In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is proposed. This algorithm treats clustering task by analyzing statistical features of data. For each data point, its values of statistical features are extracted from its neighborhood which effectively models the data proximity. By applying specifically built criteria function, TRICLUST is able to effectively handle data set with clusters of complex shapes and non-uniform densities, and with large amount of noises. One additional advantage of TRICLUST is the boundary detection function which is valuable for many real world applications such as geo-spatial data processing, point-based computer graphics, etc.

Pattern Recognition, 2008

Clustering became a classical problem in databases, data warehouses, pattern recognition, artific... more Clustering became a classical problem in databases, data warehouses, pattern recognition, artificial intelligence, and computer graphics. Applications in large spatial databases, point-based graphics, etc., give rise to new requirements for the clustering algorithms: automatic discovering of arbitrary shaped and/or non-homogeneous clusters, discovering of clusters located in low-dimensional hyperspace, detecting cluster boundaries. On that account, a new clustering and boundary detecting algorithm, ADACLUS, is proposed. It is based on the specially constructed adaptive influence function, and therefore, discovers clusters of arbitrary shapes and diverse densities, adequately captures clusters boundaries, and it is robust to noise. Normally ADACLUS performs clustering purely automatically without any preliminary parameter settings. But it also gives the user an optional possibility to set three parameters with clear meaning in order to adjust clustering for special applications. The algorithm was tested on various two-dimensional data sets, and it exhibited its effectiveness in discovering clusters of complex shapes and diverse densities. Linear complexity of the ADACLUS gives it an advantage over some well-known algorithms.

Chebyshevskii Sbornik

In the present article we propose a modification of the PaDiM anomaly detection method which maps... more In the present article we propose a modification of the PaDiM anomaly detection method which maps images to vectors and then calculates the Mahalanobis distance between such vectors and the distribution of the vectors of the training set. Of the coordinate axes of the vectors we choose a subset of such that the distribution along them is close to normal according to the chosen statistical criterion. The uniformization procedure is then applied to those coordinates and the Mahalanobis distance is calculated. This approach is shown to increase the ROCAUC value in comparison with the PaDiM method.

Integrability and Nonintegrability in Geometry and Mechanics, 1988

It isn't that they can't see the solution. It is that they can't see the problem. G.K. Chesterton... more It isn't that they can't see the solution. It is that they can't see the problem. G.K. Chesterton. The Scandal of Father Brown 'The point of a Pin' .

2007 International Conference on Cyberworlds (CW'07), 2007

ABSTRACT The mathematical structure of cyberworlds is clarified based on the duality of homology ... more ABSTRACT The mathematical structure of cyberworlds is clarified based on the duality of homology lifting property and homotopy extension property. The duality gives bottom-up and top-down methods to model, design and analyze the structure of cyberworlds. The set of homepages representing a cyberworld is transformed into a state finite machine. In development of the cyberworld, a sequence of finite state machines is obtained. This sequence has homotopic property. This property is clarified to map a finite state machine to a simplicial complex. Wikipedia, bottom-up network construction and top-down network analysis are described as examples.

Error estimate for the direct algorithm of determination of the projective mapping is presented. ... more Error estimate for the direct algorithm of determination of the projective mapping is presented. Such estimates are important in the problem of gluing of 2D projective images obtained from different points in the space [1], [2], [5]. The obtained estimate suggests that the direct algorithm of projective mapping calculation is rather accurate and robust.

Manifold hypothesis states that data points in high-dimensional space actually lie in close vicin... more Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and semi-supervised learning. Here we present new approach to manifold hypothesis checking and underlying manifold dimension estimation. In order to do it we use two very different methods simultaneously - one geometric, another probabilistic - and check whether they give the same result. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. The probabilistic method is new. Although it exploits standard nearest neighborhood distance, it is different from methods which were previously used in such situations. This method is robust, fast and includes special preliminary data transformation. Experiments on real datasets show that the suggested approach based on two methods combination...

Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight... more Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier- Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Thus, the papers published in this special issue address topics related to the Navier-Stokes equations, transition to turbulence, and some other related problems. These papers cover topics in applied mathematics, and their applications to science and engineering.

Journal | MESA, 2013

Method of Laplace transform is standard in mathematical physics. One of its advantages is that it... more Method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, Theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.

Manifold hypothesis states that data points in highdimensional space actually lie in close vicini... more Manifold hypothesis states that data points in highdimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and semi-supervised learning. Here we present new approach to manifold hypothesis checking and underlying manifold dimension estimation. In order to do it we use two very different methods simultaneously — one geometric, another probabilistic — and check whether they give the same result. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. The probabilistic method is new. Although it exploits standard nearest neighborhood distance, it is different from methods which were previously used in such situations. This method is robust, fast and includes special preliminary data transformation. Experiments on real datasets show that the suggested approach based on two methods combination ...

Formal analysis and computer recognition of 2D color images is important branch of modern compute... more Formal analysis and computer recognition of 2D color images is important branch of modern computer geometry. However, existing algorithms, although they are highly developed, are not quite satisfactory and seem to be much worse than (unknown) algorithms, which our brain uses to analyze eye information. Almost all existing algorithms omit colors and deal with grayscale transformations only. But in many cases color information is important. In this paper fundamentally new method of coding and analyzing color digital images is suggested. The main point of this method is that a full-color digital image is represented, without dropping colors, by special 2D surface in 3D space, after which it is analyzed by methods of differential geometry, rather than traditional gradient-based or Hessian-based methods (like in SIFT, GLOH, SURF, Canny operator, and many other algorithms).

Pattern Recognition, Sep 1, 2008

Clustering became a classical problem in databases, data warehouses, pattern recognition, artific... more Clustering became a classical problem in databases, data warehouses, pattern recognition, artificial intelligence, and computer graphics. Applications in large spatial databases, point-based graphics, etc., give rise to new requirements for the clustering algorithms: automatic discovering of arbitrary shaped and/or non-homogeneous clusters, discovering of clusters located in low-dimensional hyperspace, detecting cluster boundaries. On that account, a new clustering and boundary detecting algorithm, ADACLUS, is proposed. It is based on the specially constructed adaptive influence function, and therefore, discovers clusters of arbitrary shapes and diverse densities, adequately captures clusters boundaries, and it is robust to noise. Normally ADACLUS performs clustering purely automatically without any preliminary parameter settings. But it also gives the user an optional possibility to set three parameters with clear meaning in order to adjust clustering for special applications. The algorithm was tested on various two-dimensional data sets, and it exhibited its effectiveness in discovering clusters of complex shapes and diverse densities. Linear complexity of the ADACLUS gives it an advantage over some well-known algorithms.

Pattern Recognition Letters, Jul 1, 2008

In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is pro... more In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is proposed. This algorithm treats clustering task by analyzing statistical features of data. For each data point, its values of statistical features are extracted from its neighborhood which effectively models the data proximity. By applying specifically built criteria function, TRICLUST is able to effectively handle data set with clusters of complex shapes and non-uniform densities, and with large amount of noises. One additional advantage of TRICLUST is the boundary detection function which is valuable for many real world applications such as geo-spatial data processing, point-based computer graphics, etc.

A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new c... more A novel mathematical model of micro socio-economic behavior is proposed. It is based on the new concept of the informational channel, viewed as an elementary socio-economic cell. The processes, taking place within this cell, are described by a set of partial differential equations that relate the two main aspects of a socio-economic system: energy (any material asset, such as money, material property, etc.) and information. The solution is obtained in the form of the Volterra integral equation that relates the value of disturbance upon the socio-economical cell and the value of the wealth function, which is defined as an integral characteristic of both material and informational components. Two basic cases of disturbance were analyzed: a constant and a single-pulse flux. In both cases, the results show a quite natural behavior. The proposed model might serve as a basis for more advanced analysis of socio-economical systems.

From the text: Waves follow our boat as we meander across the lake, and turbulent air currents fo... more From the text: Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. Thus, the papers published in this special issue address topics related to the Navier-Stokes equations, transition to turbulence, and some other related problems. These papers cover topics in applied mathematics, and their applications to science and engineering.

The method of Laplace transform is standard in mathematical physics. One of its advantages is tha... more The method of Laplace transform is standard in mathematical physics. One of its advantages is that it is still applicable in the case of PDE with fractional derivatives. But the situation becomes much more complicated if the rate of differentiation can vary with time, even if these variations have very small amplitude. In this paper, a theorem is proved which gives the expression for the Laplace transform of the diffusion equation with time-dependent rate of time derivative. Such equations appear in biology and financial mathematics. It follows that in case of variable fractional derivatives the method of Laplace transform can be used only under certain conditions as approximate method.

2007 International Conference on Cyberworlds (CW'07), 2007

Pattern Recognition Letters, 2008

In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is pro... more In this paper, a new spatial clustering algorithm TRICLUST based on Delaunay triangulation is proposed. This algorithm treats clustering task by analyzing statistical features of data. For each data point, its values of statistical features are extracted from its neighborhood which effectively models the data proximity. By applying specifically built criteria function, TRICLUST is able to effectively handle data set with clusters of complex shapes and non-uniform densities, and with large amount of noises. One additional advantage of TRICLUST is the boundary detection function which is valuable for many real world applications such as geo-spatial data processing, point-based computer graphics, etc.

Pattern Recognition, 2008

Clustering became a classical problem in databases, data warehouses, pattern recognition, artific... more Clustering became a classical problem in databases, data warehouses, pattern recognition, artificial intelligence, and computer graphics. Applications in large spatial databases, point-based graphics, etc., give rise to new requirements for the clustering algorithms: automatic discovering of arbitrary shaped and/or non-homogeneous clusters, discovering of clusters located in low-dimensional hyperspace, detecting cluster boundaries. On that account, a new clustering and boundary detecting algorithm, ADACLUS, is proposed. It is based on the specially constructed adaptive influence function, and therefore, discovers clusters of arbitrary shapes and diverse densities, adequately captures clusters boundaries, and it is robust to noise. Normally ADACLUS performs clustering purely automatically without any preliminary parameter settings. But it also gives the user an optional possibility to set three parameters with clear meaning in order to adjust clustering for special applications. The algorithm was tested on various two-dimensional data sets, and it exhibited its effectiveness in discovering clusters of complex shapes and diverse densities. Linear complexity of the ADACLUS gives it an advantage over some well-known algorithms.