Randomized Dynamic Mode Decomposition

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum

Efficient representations of large radiosity matrices

The radiosity equation can be expressed as a linear system, where light interactions between patches of the scene are considered. Its resolution has been one of the main subjects in computer graphics, which has lead to the development of methods focused on different goals. For instance, in inverse lighting problems, it is convenient to solve the radiosity equation thousands of times for static geometries. Also, this calculation needs to consider many (or infinite) light bounces to achieve accurate global illumination results. Several methods have been developed to solve the linear system by finding approximations or other representations of the radiosity matrix, because the full storage of this matrix is memory demanding. Some examples are hierarchical radiosity, progressive refinement approaches, or wavelet radiosity. Even though these methods are memory efficient, they may become slow for many light bounces, due to their iterative nature. Recently, efficient methods have been developed for the direct resolution of the radiosity equation. In this case, the challenge is to reduce the memory requirements of the radiosity matrix, and its inverse. The main objective of this thesis is exploiting the properties of specific problems to reduce the memory requirements of the radiosity problem. Hereby, two types of problems are analyzed. The first problem is to solve radiosity for scenes with a high spatial coherence, such as it happens to some architectural models. The second involves scenes with a high occlusion factor between patches. For the high spatial coherence case, a novel and efficient error-bounded factorization method is presented. It is based on the use of multiple singular value decompositions along with a space filling curve, which allows to exploit spatial coherence. This technique accelerates the factorization of in-core matrices, and allows to work with out-of-core matrices passing only one time over them. In the experimental analysis, the presented method is applied to scenes up to 163K patches. After a precomputation stage, it is used to solve the radiosity equation for fixed geometries and infinite bounces, at interactive times. For the high occlusion problem, city models are used. In this case, the sparsity of the radiosity matrix is exploited. An approach for radiative exchange computation is proposed, where the inverse of the radiosity matrix is approximated. In this calculation, near-zero elements are removed, leading to a highly sparse result. This technique is applied to simulate daylight in urban environments composed by up to 140k patches.La ecuación de radiosidad tiene por objetivo el cálculo de la interacción de la luz con los elementos de la escena. Esta se puede expresar como un sistema lineal, cuya resolución ha derivado en el desarrollo de diversos métodos gráficos para satisfacer propósitos específicos. Por ejemplo, en problemas inversos de iluminación para geometrías estáticas, se debe resolver la ecuación de radiosidad miles de veces. Además, este cálculo debe considerar muchos (infinitos) rebotes de luz, si se quieren obtener resultados precisos de iluminación global. Entre los métodos desarrollados, se destacan aquellos que generan aproximaciones u otras representaciones de la matriz de radiosidad, debido a que su almacenamiento requiere grandes cantidades de memoria. Algunos ejemplos de estas técnicas son la radiosidad jerárquica, el refinamiento progresivo y la radiosidad basada en wavelets. Si bien estos métodos son eficientes en cuanto a memoria, pueden ser lentos cuando se requiere el cálculo de muchos rebotes de luz, debido a su naturaleza iterativa. Recientemente se han desarrollado métodos eficientes para la resolución directa de la ecuación de radiosidad, basados en el pre-cómputo de la inversa de la matriz de radiosidad. En estos casos, el desafío consiste en reducir los requerimientos de memoria y tiempo de ejecución para el cálculo de la matriz y de su inversa. El principal objetivo de la tesis consiste en explotar propiedades específicas de ciertos problemas de iluminación para reducir los requerimientos de memoria de la ecuación de radiosidad. En este contexto, se analizan dos casos diferentes. El primero consiste en hallar la radiosidad para escenas con alta coherencia espacial, tal como ocurre en algunos modelos arquitectónicos. El segundo involucra escenas con un elevado factor de oclusión entre parches. Para el caso de alta coherencia espacial, se presenta un nuevo método de factorización de matrices que es computacionalmente eficiente y que genera aproximaciones cuyo error es configurable. Está basado en el uso de múltiples descomposiciones en valores singulares (SVD) junto a una curva de recubrimiento espacial, lo que permite explotar la coherencia espacial. Esta técnica acelera la factorización de matrices que entran en memoria, y permite trabajar con matrices que no entran en memoria, recorriéndolas una única vez. En el análisis experimental, el método presentado es aplicado a escenas de hasta 163 mil parches. Luego de una etapa de precómputo, se logra resolver la ecuación de radiosidad en tiempos interactivos, para geométricas estáticas e infinitos rebotes. Para el problema de alta oclusión, se utilizan modelos de ciudades. En este caso, se aprovecha la baja densidad de la matriz de radiosidad, y se propone una técnica para el cálculo aproximado de su inversa. En este cálculo, los elementos cercanos a cero son eliminados. La técnica es aplicada a la simulación de la luz natural en ambientes urbanos compuestos por hasta 140 mil parches

Time Efficiency on Computational Performance of PCA, FA and TSVD on Ransomware Detection

Ransomware is able to attack and take over access of the targeted user'scomputer. Then the hackers demand a ransom to restore the user's accessrights. Ransomware detection process especially in big data has problems interm of computational processing time or detection speed. Thus, it requires adimensionality reduction method for computational process efficiency. Thisresearch work investigates the efficiency of three dimensionality reductionmethods, i.e.: Principal Component Analysis (PCA), Factor Analysis (FA) andTruncated Singular Value Decomposition (TSVD). Experimental results onCICAndMal2017 dataset show that PCA is the fastest and most significantmethod in the computational process with average detection time of 34.33s.Furthermore, result of accuracy, precision and recall also show that the PCAis superior compared to FA and TSVD

Unsupervised Adaptation for High-Dimensional with Limited-Sample Data Classification Using Variational Autoencoder

High-dimensional with limited-sample size (HDLSS) datasets exhibit two critical problems: (1) Due to the insufficiently small-sample size, there is a lack of enough samples to build classification models. Classification models with a limited-sample may lead to overfitting and produce erroneous or meaningless results. (2) The 'curse of dimensionality' phenomena is often an obstacle to the use of many methods for solving the high-dimensional with limited-sample size problem and reduces classification accuracy. This study proposes an unsupervised framework for high-dimensional limited-sample size data classification using dimension reduction based on variational autoencoder (VAE). First, the deep learning method variational autoencoder is applied to project high-dimensional data onto lower-dimensional space. Then, clustering is applied to the obtained latent-space of VAE to find the data groups and classify input data. The method is validated by comparing the clustering results with actual labels using purity, rand index, and normalized mutual information. Moreover, to evaluate the proposed model strength, we analyzed 14 datasets from the Arizona State University Digital Repository. Also, an empirical comparison of dimensionality reduction techniques shown to conclude their applicability in the high-dimensional with limited-sample size data settings. Experimental results demonstrate that variational autoencoder can achieve more accuracy than traditional dimensionality reduction techniques in high-dimensional with limited-sample-size data analysis

Time efficiency on computational performance of PCA, FA and TSVD on ransomware detection

Ransomware is able to attack and take over access of the targeted user's computer. Then the hackers demand a ransom to restore the user's access rights. Ransomware detection process especially in big data has problems in term of computational processing time or detection speed. Thus, it requires a dimensionality reduction method for computational process efficiency. This research work investigates the efficiency of three dimensionality reduction methods, i.e.: Principal Component Analysis (PCA), Factor Analysis (FA) and Truncated Singular Value Decomposition (TSVD). Experimental results on CICAndMal2017 dataset show that PCA is the fastest and most significant method in the computational process with average detection time of 34.33s. Furthermore, result of accuracy, precision and recall also show that the PCA is superior compared to FA and TSVD