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Ivan Oseledets

Publications and source records attributed to Ivan Oseledets.

At least 217 records · Page 12Linked to original sources
arXiv2018

Modelling hidden structure of signals in group data analysis with modified (Lr, 1) and block-term decompositions

This work is devoted to elaboration on the idea to use block term decomposition for group data analysis and to raise the possibility of modelling group activity with (Lr, 1) and Tucker blocks. A new generalization of block tensor decomposition was considered in application to group data analysis. Suggested approach was evaluated on multilabel classification task for a set of images. This contribution also reports results of investigation on clustering with proposed tensor models in comparison with known matrix models, namely common orthogonal basis extraction and group independent component analysis.

math.NA↗
arXiv2018

Revealing the Unobserved by Linking Collaborative Behavior and Side Knowledge

We propose a tensor-based model that fuses a more granular representation of user preferences with the ability to take additional side information into account. The model relies on the concept of ordinal nature of utility, which better corresponds to actual user perception. In addition to that, unlike the majority of hybrid recommenders, the model ties side information directly to collaborative data, which not only addresses the problem of extreme data sparsity, but also allows to naturally exploit patterns in the observed behavior for a more meaningful representation of user intents. We demonstrate the effectiveness of the proposed model on several standard benchmark datasets. The general formulation of the approach imposes no restrictions on the type of observed interactions and makes it potentially applicable for joint modelling of context information along with side data.

cs.IR↗
arXiv2018

Comparative study of Discrete Wavelet Transforms and Wavelet Tensor Train decomposition to feature extraction of FTIR data of medicinal plants

Fourier-transform infra-red (FTIR) spectra of samples from 7 plant species were used to explore the influence of preprocessing and feature extraction on efficiency of machine learning algorithms. Wavelet Tensor Train (WTT) and Discrete Wavelet Transforms (DWT) were compared as feature extraction techniques for FTIR data of medicinal plants. Various combinations of signal processing steps showed different behavior when applied to classification and clustering tasks. Best results for WTT and DWT found through grid search were similar, significantly improving quality of clustering as well as classification accuracy for tuned logistic regression in comparison to original spectra. Unlike DWT, WTT has only one parameter to be tuned (rank), making it a more versatile and easier to use as a data processing tool in various signal processing applications.

eess.SP↗
arXiv2018

Geometry Score: A Method For Comparing Generative Adversarial Networks

One of the biggest challenges in the research of generative adversarial networks (GANs) is assessing the quality of generated samples and detecting various levels of mode collapse. In this work, we construct a novel measure of performance of a GAN by comparing geometrical properties of the underlying data manifold and the generated one, which provides both qualitative and quantitative means for evaluation. Our algorithm can be applied to datasets of an arbitrary nature and is not limited to visual data. We test the obtained metric on various real-life models and datasets and demonstrate that our method provides new insights into properties of GANs.

cs.LG↗
arXiv2018

Simple non-extensive sparsification of the hierarchical matrices

In this paper, we consider the matrices approximated in H2 format. The direct solution, as well as the preconditioning, of systems with such matrices is a challenging problem. We propose a non-extensive sparse factorization of the H2 matrix that allows to substitute the direct H2 solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed out of parameters of the H2 matrix. In the numerical experiments, we show the consistency of this approach in comparison to the other approximate block low-rank hierarchical solvers, such as HODLR, H2Lib and IFMM.

math.NA↗
arXiv2018

Regularization of topology optimization problem by the FEM a posteriori error estimator

In our work, we consider the classical density-based approach to topology optimization. We propose the modification of the discretized cost/objective functional using a posteriori error estimator for the finite element method. It can be regarded as a new technique to prevent checkerboards. It also provides higher regularity of the solutions and robustness of the results.

math.NA↗
arXiv2018

Tensor Methods and Recommender Systems

A substantial progress in development of new and efficient tensor factorization techniques has led to an extensive research of their applicability in recommender systems field. Tensor-based recommender models push the boundaries of traditional collaborative filtering techniques by taking into account a multifaceted nature of real environments, which allows to produce more accurate, situational (e.g. context-aware, criteria-driven) recommendations. Despite the promising results, tensor-based methods are poorly covered in existing recommender systems surveys. This survey aims to complement previous works and provide a comprehensive overview on the subject. To the best of our knowledge, this is the first attempt to consolidate studies from various application domains in an easily readable, digestible format, which helps to get a notion of the current state of the field. We also provide a high level discussion of the future perspectives and directions for further improvement of tensor-based recommendation systems.

cs.LG↗
arXiv2018

Expressive power of recurrent neural networks

Deep neural networks are surprisingly efficient at solving practical tasks, but the theory behind this phenomenon is only starting to catch up with the practice. Numerous works show that depth is the key to this efficiency. A certain class of deep convolutional networks -- namely those that correspond to the Hierarchical Tucker (HT) tensor decomposition -- has been proven to have exponentially higher expressive power than shallow networks. I.e. a shallow network of exponential width is required to realize the same score function as computed by the deep architecture. In this paper, we prove the expressive power theorem (an exponential lower bound on the width of the equivalent shallow network) for a class of recurrent neural networks -- ones that correspond to the Tensor Train (TT) decomposition. This means that even processing an image patch by patch with an RNN can be exponentially more efficient than a (shallow) convolutional network with one hidden layer. Using theoretical results on the relation between the tensor decompositions we compare expressive powers of the HT- and TT-Networks. We also implement the recurrent TT-Networks and provide numerical evidence of their expressivity.

cs.LG↗
arXiv2018

Function approximation using gradient information with application to parametric and stochastic differential equations

In the paper we consider the problem of multivariate function approximation in polynomial basis. In order to solve this problem, we adjust the least squares method (LSM) by adding information about derivatives of the function. This modification allows reducing the number of evaluations of approximating function while keeping the accuracy at the appropriate level. We propose several techniques for time-efficient calculation of derivatives in various applications. Numerical examples are given for comparison between the standard LSM and the proposed approach.

math.NA↗
arXiv2017

Exponential Machines

Modeling interactions between features improves the performance of machine learning solutions in many domains (e.g. recommender systems or sentiment analysis). In this paper, we introduce Exponential Machines (ExM), a predictor that models all interactions of every order. The key idea is to represent an exponentially large tensor of parameters in a factorized format called Tensor Train (TT). The Tensor Train format regularizes the model and lets you control the number of underlying parameters. To train the model, we develop a stochastic Riemannian optimization procedure, which allows us to fit tensors with 2^160 entries. We show that the model achieves state-of-the-art performance on synthetic data with high-order interactions and that it works on par with high-order factorization machines on a recommender system dataset MovieLens 100K.

stat.ML↗
arXiv2017

Art of singular vectors and universal adversarial perturbations

Vulnerability of Deep Neural Networks (DNNs) to adversarial attacks has been attracting a lot of attention in recent studies. It has been shown that for many state of the art DNNs performing image classification there exist universal adversarial perturbations --- image-agnostic perturbations mere addition of which to natural images with high probability leads to their misclassification. In this work we propose a new algorithm for constructing such universal perturbations. Our approach is based on computing the so-called $(p, q)$-singular vectors of the Jacobian matrices of hidden layers of a network. Resulting perturbations present interesting visual patterns, and by using only 64 images we were able to construct universal perturbations with more than 60 \% fooling rate on the dataset consisting of 50000 images. We also investigate a correlation between the maximal singular value of the Jacobian matrix and the fooling rate of the corresponding singular vector, and show that the constructed perturbations generalize across networks.

cs.CV↗
arXiv2017

Deep Multigrid: learning prolongation and restriction matrices

This paper proposes the method to optimize restriction and prolongation operators in the two-grid method. The proposed method is straightforwardly extended to the geometric multigrid method (GMM). GMM is used in solving discretized partial differential equation (PDE) and based on the restriction and prolongation operators. The operators are crucial for fast convergence of GMM, but they are unknown. To find them we propose a reformulation of the two-grid method in terms of a deep neural network with a specific architecture. This architecture is based on the idea that every operation in the two-grid method can be considered as a layer of a deep neural network. The parameters of layers correspond to the restriction and prolongation operators. Therefore, we state an optimization problem with respect to these operators and get optimal ones through backpropagation approach. To illustrate the performance of the proposed approach, we carry out experiments on the discretized Laplace equation, Helmholtz equation and singularly perturbed convection-diffusion equation and demonstrate that proposed approach gives operators, which lead to faster convergence.

math.NA↗
arXiv2017

Desingularization of bounded-rank matrix sets

Conventional ways to solve optimization problems on low-rank matrix sets which appear in great number of applications ignore its underlying structure of an algebraic variety and existence of singular points. This leads to appearance of inverses of singular values in algorithms and since they could be close to $0$ it causes certain problems. We tackle this problem by utilizing ideas from the algebraic geometry and show how to desingularize these sets. Our main result is algorithm which uses only bounded functions of singular values and hence does not suffer from the issue described above.

math.NA↗
arXiv2017

Neural networks for topology optimization

In this research, we propose a deep learning based approach for speeding up the topology optimization methods. The problem we seek to solve is the layout problem. The main novelty of this work is to state the problem as an image segmentation task. We leverage the power of deep learning methods as the efficient pixel-wise image labeling technique to perform the topology optimization. We introduce convolutional encoder-decoder architecture and the overall approach of solving the above-described problem with high performance. The conducted experiments demonstrate the significant acceleration of the optimization process. The proposed approach has excellent generalization properties. We demonstrate the ability of the application of the proposed model to other problems. The successful results, as well as the drawbacks of the current method, are discussed.

cs.LG↗
arXiv2017

Time- and memory-efficient representation of complex mesoscale potentials

We apply the modern technique of approximation of multivariate functions - tensor train cross approximation - to the problem of the description of physical interactions between complex-shaped bodies in a context of computational nanomechanics. In this note we showcase one particular example - van der Waals interactions between two cylindrical bodies - relevant to modeling of carbon nanotube systems. The potential is viewed as a tensor (multidimensional table) which is represented in compact form with the help of tensor train decomposition. The described approach offers a universal solution for the description of van der Waals interactions between complex-shaped nanostructures and can be used within the framework of such systems of mesoscale modeling as recently emerged mesoscopic distinct element method (MDEM).

physics.comp-ph↗
arXiv2017

Riemannian Optimization for Skip-Gram Negative Sampling

Skip-Gram Negative Sampling (SGNS) word embedding model, well known by its implementation in "word2vec" software, is usually optimized by stochastic gradient descent. However, the optimization of SGNS objective can be viewed as a problem of searching for a good matrix with the low-rank constraint. The most standard way to solve this type of problems is to apply Riemannian optimization framework to optimize the SGNS objective over the manifold of required low-rank matrices. In this paper, we propose an algorithm that optimizes SGNS objective using Riemannian optimization and demonstrates its superiority over popular competitors, such as the original method to train SGNS and SVD over SPPMI matrix.

cs.CL↗
arXiv2017

Vico-Greengard-Ferrando quadratures in the tensor solver for integral equations

Convolution with Green's function of a differential operator appears in a lot of applications e.g. Lippmann-Schwinger integral equation. Algorithms for computing such are usually non-trivial and require non-uniform mesh. However, recently Vico, Greengard and Ferrando developed method for computing convolution with smooth functions with compact support with spectral accuracy, requiring nothing more than Fast Fourier Transform (FFT). Their approach is very suitable for the low-rank tensor implementation which we develop using Quantized Tensor Train (QTT) decomposition.

math.NA↗