Interactive certificate for the verification of Wiedemann's Krylov sequence: application to the certification of the determinant, the minimal and the characteristic polynomials of sparse matrices

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a-possibly randomized- verification algorithm that proves the correctness of each output. Wiede-mann's algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divides the minimal polynomial of the matrix. For instance, if the $n\times n$ input matrix is sparse with n 1+o(1) non-zero entries, the computation of the sequence is quadratic in the dimension of the matrix while the computation of the minimal polynomial is n 1+o(1), once that projected Krylov sequence is obtained. In this paper we give algorithms that compute certificates for the Krylov sequence of sparse or structured $n\times n$ matrices over an abstract field, whose Monte Carlo verification complexity can be made essentially linear. As an application this gives certificates for the determinant, the minimal and characteristic polynomials of sparse or structured matrices at the same cost

Tehisnärvivõrgud bioloogiliste andmete analüüsimiseks

Väitekirja elektrooniline versioon ei sisalda publikatsiooneTehisnärvivõrgud viimastel aastatel populaarsust kogunud masinõppe algoritm, mis on võimeline näidete põhjal õppima. Erinevad tehisnärvivõrkude alamtüübid on kasutusel mitmetes arvutiteaduse harudes: konvolutsioonilisi võrke rakendatakse objekti- ja näotuvastuses; rekurrentsed võrgud on efektiivsed kõnetuvastuses ja keeletehnoloogias. Need ei ole aga ainsad võimalikud tehisnärvivõrkude rakendamise valdkonnad - selles doktoritöös näitasime me tehisnärvivõrkude kasulikkust kahe bioloogilise probleemi lahendamisel. Esiteks küsisime, kas ainult DNA jupis sisalduva info põhjal on võimalik ennustada, kas see järjestus pärineb viiruse (ja mitte mõnda muud tüüpi organismi) genoomist. Läbi kahe publikatsiooni tõestasime me, et masinõppe algoritmid on selleks tõesti võimelised. Parima täpsuse saavutas konvolutsiooniline närvivõrk. Loodud lahendus võimaldab viroloogidel tuvastada seni tundmatuid viiruseliike, millel võib olla oluline mõju inimese tervisele. Teine käsitletud bioloogiline andmestik pärineb neuroteadusest. Imetajate hipokampuses esineb nn. koharakke, mis aktiveeruvad vaid juhul, kui loom asub teatud ruumipunktis. Näitasime, et rekurrentsete närvivõrkude abil saab vaid mõnekümne koharaku aktiivsuse põhjal ennustada roti asukohta ligi 10 cm täpsusega. Rekurrentsed võrgud osutusid efektiivsemaks kui neuroteaduses enim levinud Bayesi meetodid. Need võrgud suudavad kasutada rakkude eelnevat aktiivsust kontekstina, mis aitab täpsustada asukoha ennustust. Ka teistes neuroandmestikes võib eelnev ajuaktiivsus peegeldada konteksti, mis sisaldab olulist infot hetkel toimuva kohta. Seega võivad rekurrentsed tehisnärvivõrgud osutuda ajusignaalide mõistmisel ülimalt kasulikuks. Samuti on bioinformaatikas veel hulk andmestikke, kus konvolutsioonilised võrgud võivad osutuda efektiivsemaks kui senised meetodid. Loodame, et käesolev töö julgustab teadlasi tehisnärvivõrke proovima ka oma andmestikel.Artificial neural networks (ANNs) are a machine learning algorithm that has gained popularity in recent years. Different subtypes of ANNs are used in various fields of computer science. For example, convolutional networks are useful in object and face recognition systems; whereas recurrent neural networks are effective in speech recognition and natural language processing. However, these examples are not the only possible applications of neural nets - in this thesis we demonstrated the benefits of ANNs in analyzing two biological datasets. First, we investigated if based only on the information contained within a DNA snippet it is possible to predict if the snippet originates from a viral genome or not. Through two publications we demonstrated that machine learning algorithms can make this prediction. Convolutional neural networks (CNNs) proved to be the most accurate. The recommendation system created allows virologists to identify yet unknown viral species, which may have important effects on human health. The second biological dataset analyzed originates from neuroscience. In mammalian hippocampus there are so called place cells which activate only if the animal is in a specific location in space. We showed that recurrent neural networks (RNNs) allow to predict the animal’s location with ~10cm precision based on the activity of only a few dozen place cells. RNNs proved to be more effective than the most commonly used Bayesian methods. These networks use the past neuronal activity as a context that helps fine-tune the location predictions. Also in many other neural datasets the prior brain activity might reflect important information about the current behaviour. Hence, RNNs might turn out to be very useful in making sense of brain signals. Similarly, CNNs are likely to prove more efficient than the currently used methods on many other bioinformatics datasets. We hope this thesis encourages more scientists to try neural networks on their own datasets.https://www.ester.ee/record=b536839

Fast Computation of Special Resultants

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series

DEEP LEARNING TECHNIQUES FOR DETECTION OF FALSE DATA INJECTION ATTACKS ON ELECTRIC POWER GRID

The electric power grid uses a set of measuring and switching devices for its operations and control. The data retrieved from the measuring instruments is assumed to be noisy, therefore a state estimator is used to estimate the correct values of state variables on which the system can take control actions. The modern electric power grid is dependent on communication networks for transferring these measurements, which are susceptible to intrusions from hackers. False data injection attacks (FDIA) are one of the most common attack strategies where an intruder tries to trick the underlying control system of the grid to cause disruptions without getting detected by native anomaly detection measures inbuilt in the state estimator. The native anomaly detection mechanism relies on threshold and residual based measure to flag a set of measurements as anomaly. Therefore, if the attack is devised in such a way that the intrusion can be performed without significantly affecting the residual error of state estimation it can go undetected. We propose a data augmented deep learning based solution to detect such attacks in real time. We propose methods of generating realistic random and targeted attack simulations on standard IEEE architectures and methods of detecting them using deep learning models. We propose recurrent neural network (RNN) based architectures to detect and locate FDIAs and devices compromised in real-time. For detection we propose a supervised and an unsupervised method. Similarly, for location we propose a method to find exact devices compromised which is less practical and then move on to a more feasible and practical solution in supervised and unsupervised conditions. Being an intrusion detection system it is critical to detect all attacks which means false negatives should be penalized heavily, whereas false positives can be accommodated. Therefore, we use recall as our primary performance metric and precision recall curve to find an optimal threshold of probability score. In addition, we demonstrate how our approach is better than a residual error and other previous detection models. We also compare the performance of our models with increasing number of devices being compromised

Secure Outsourced Computation of the Characteristic Polynomial and Eigenvalues of Matrix

Linear algebra plays an important role in computer science, especially in cryptography.Numerous cryptog-raphic protocols, scientific computations, and numerical computations are based on linear algebra. Many linear algebra tasks can be reduced to some core problems, such as matrix multiplication, determinant of matrix and the characteristic polynomial of matrix. However, it is difficult to execute these tasks independently for client whose computation abilities are weaker than polynomial-time computational ability. Cloud Computing is a novel economical paradigm which provides powerful computational resources that enables resources-constrained client to outsource their mass computing tasks to the cloud. In this paper, we propose a new verifiable and secure outsourcing protocol for the problem of computing the characteristic polynomial and eigenvalues of matrix. These protocols are not only efficient and secure, but also unnecessary for any cryptographic assumption

Spatiotemporal anomaly detection: streaming architecture and algorithms

Includes bibliographical references.2020 Summer.Anomaly detection is the science of identifying one or more rare or unexplainable samples or events in a dataset or data stream. The field of anomaly detection has been extensively studied by mathematicians, statisticians, economists, engineers, and computer scientists. One open research question remains the design of distributed cloud-based architectures and algorithms that can accurately identify anomalies in previously unseen, unlabeled streaming, multivariate spatiotemporal data. With streaming data, time is of the essence, and insights are perishable. Real-world streaming spatiotemporal data originate from many sources, including mobile phones, supervisory control and data acquisition enabled (SCADA) devices, the internet-of-things (IoT), distributed sensor networks, and social media. Baseline experiments are performed on four (4) non-streaming, static anomaly detection multivariate datasets using unsupervised offline traditional machine learning (TML), and unsupervised neural network techniques. Multiple architectures, including autoencoders, generative adversarial networks, convolutional networks, and recurrent networks, are adapted for experimentation. Extensive experimentation demonstrates that neural networks produce superior detection accuracy over TML techniques. These same neural network architectures can be extended to process unlabeled spatiotemporal streaming using online learning. Space and time relationships are further exploited to provide additional insights and increased anomaly detection accuracy. A novel domain-independent architecture and set of algorithms called the Spatiotemporal Anomaly Detection Environment (STADE) is formulated. STADE is based on federated learning architecture. STADE streaming algorithms are based on a geographically unique, persistently executing neural networks using online stochastic gradient descent (SGD). STADE is designed to be pluggable, meaning that alternative algorithms may be substituted or combined to form an ensemble. STADE incorporates a Stream Anomaly Detector (SAD) and a Federated Anomaly Detector (FAD). The SAD executes at multiple locations on streaming data, while the FAD executes at a single server and identifies global patterns and relationships among the site anomalies. Each STADE site streams anomaly scores to the centralized FAD server for further spatiotemporal dependency analysis and logging. The FAD is based on recent advances in DNN-based federated learning. A STADE testbed is implemented to facilitate globally distributed experimentation using low-cost, commercial cloud infrastructure provided by Microsoft™. STADE testbed sites are situated in the cloud within each continent: Africa, Asia, Australia, Europe, North America, and South America. Communication occurs over the commercial internet. Three STADE case studies are investigated. The first case study processes commercial air traffic flows, the second case study processes global earthquake measurements, and the third case study processes social media (i.e., Twitter™) feeds. These case studies confirm that STADE is a viable architecture for the near real-time identification of anomalies in streaming data originating from (possibly) computationally disadvantaged, geographically dispersed sites. Moreover, the addition of the FAD provides enhanced anomaly detection capability. Since STADE is domain-independent, these findings can be easily extended to additional application domains and use cases

Fast simulation of large-scale growth models

We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a "least action principle" which characterizes the odometer function of the growth process. Starting from an approximation for the odometer, we successively correct under- and overestimates and provably arrive at the correct final state. Internal diffusion-limited aggregation (IDLA) is one of the models amenable to our technique. The boundary fluctuations in IDLA were recently proved to be at most logarithmic in the size of the growth cluster, but the constant in front of the logarithm is still not known. As an application of our method, we calculate the size of fluctuations over two orders of magnitude beyond previous simulations, and use the results to estimate this constant.Comment: 27 pages, 9 figures. To appear in Random Structures & Algorithm

Constructing networks of quantum channels for state preparation

Entangled possibly mixed states are an essential resource for quantum computation, communication, metrology, and the simulation of many-body systems. It is important to develop and improve preparation protocols for such states. One possible way to prepare states of interest is to design an open system that evolves only towards the desired states. A Markovian evolution of a quantum system can be generally described by a Lindbladian. Tensor networks provide a framework to construct physically relevant entangled states. In particular, matrix product density operators (MPDOs) form an important variational class of states. MPDOs generalize matrix product states to mixed states, can represent thermal states of local one-dimensional Hamiltonians at sufficiently large temperatures, describe systems that satisfy the area law of entanglement, and form the basis of powerful numerical methods. In this work we develop an algorithm that determines for a given linear subspace of MPDOs whether this subspace can be the stable space of some frustration free k-local Lindbladian and, if so, outputs an appropriate Lindbladian. We proceed by using machine learning with networks of quantum channels, also known as quantum neural networks (QNNs), to train denoising post-processing devices for quantum sources. First, we show that QNNs can be trained on imperfect devices even when part of the training data is corrupted. Second, we show that QNNs can be trained to extrapolate quantum states to, e.g., lower temperatures. Third, we show how to denoise quantum states in an unsupervised manner. We develop a novel quantum autoencoder that successfully denoises Greenberger-Horne-Zeilinger, W, Dicke, and cluster states subject to spin-flip, dephasing errors, and random unitary noise. Finally, we develop recurrent QNNs (RQNNs) for denoising that requires memory, such as combating drifts. RQNNs can be thought of as matrix product quantum channels with a quantum algorithm for training and are closely related to MPDOs. The proposed preparation and denoising protocols can be beneficial for various emergent quantum technologies and are within reach of present-day experiments