Faster Matrix Multiplication via Asymmetric Hashing

Fast matrix multiplication is one of the most fundamental problems in algorithm research. The exponent of the optimal time complexity of matrix multiplication is usually denoted by $\omega$. This paper discusses new ideas for improving the laser method for fast matrix multiplication. We observe that the analysis of higher powers of the Coppersmith-Winograd tensor [Coppersmith & Winograd 1990] incurs a "combination loss", and we partially compensate for it using an asymmetric version of CW's hashing method. By analyzing the eighth power of the CW tensor, we give a new bound of $\omega<2.37188$, which improves the previous best bound of $\omega<2.37286$ [Alman & Vassilevska Williams 2020]. Our result breaks the lower bound of $2.3725$ in [Ambainis, Filmus & Le Gall 2015] because of the new method for analyzing component (constituent) tensors.Comment: 67 page

Tensor-based regression models and applications

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2017-2018Avec l’avancement des technologies modernes, les tenseurs d’ordre élevé sont assez répandus et abondent dans un large éventail d’applications telles que la neuroscience informatique, la vision par ordinateur, le traitement du signal et ainsi de suite. La principale raison pour laquelle les méthodes de régression classiques ne parviennent pas à traiter de façon appropriée des tenseurs d’ordre élevé est due au fait que ces données contiennent des informations structurelles multi-voies qui ne peuvent pas être capturées directement par les modèles conventionnels de régression vectorielle ou matricielle. En outre, la très grande dimensionnalité de l’entrée tensorielle produit une énorme quantité de paramètres, ce qui rompt les garanties théoriques des approches de régression classique. De plus, les modèles classiques de régression se sont avérés limités en termes de difficulté d’interprétation, de sensibilité au bruit et d’absence d’unicité. Pour faire face à ces défis, nous étudions une nouvelle classe de modèles de régression, appelés modèles de régression tensor-variable, où les prédicteurs indépendants et (ou) les réponses dépendantes prennent la forme de représentations tensorielles d’ordre élevé. Nous les appliquons également dans de nombreuses applications du monde réel pour vérifier leur efficacité et leur efficacité.With the advancement of modern technologies, high-order tensors are quite widespread and abound in a broad range of applications such as computational neuroscience, computer vision, signal processing and so on. The primary reason that classical regression methods fail to appropriately handle high-order tensors is due to the fact that those data contain multiway structural information which cannot be directly captured by the conventional vector-based or matrix-based regression models, causing substantial information loss during the regression. Furthermore, the ultrahigh dimensionality of tensorial input produces huge amount of parameters, which breaks the theoretical guarantees of classical regression approaches. Additionally, the classical regression models have also been shown to be limited in terms of difficulty of interpretation, sensitivity to noise and absence of uniqueness. To deal with these challenges, we investigate a novel class of regression models, called tensorvariate regression models, where the independent predictors and (or) dependent responses take the form of high-order tensorial representations. We also apply them in numerous real-world applications to verify their efficiency and effectiveness. Concretely, we first introduce hierarchical Tucker tensor regression, a generalized linear tensor regression model that is able to handle potentially much higher order tensor input. Then, we work on online local Gaussian process for tensor-variate regression, an efficient nonlinear GPbased approach that can process large data sets at constant time in a sequential way. Next, we present a computationally efficient online tensor regression algorithm with general tensorial input and output, called incremental higher-order partial least squares, for the setting of infinite time-dependent tensor streams. Thereafter, we propose a super-fast sequential tensor regression framework for general tensor sequences, namely recursive higher-order partial least squares, which addresses issues of limited storage space and fast processing time allowed by dynamic environments. Finally, we introduce kernel-based multiblock tensor partial least squares, a new generalized nonlinear framework that is capable of predicting a set of tensor blocks by merging a set of tensor blocks from different sources with a boosted predictive power

Robot Learning from Human Demonstration: Interpretation, Adaptation, and Interaction

Robot Learning from Demonstration (LfD) is a research area that focuses on how robots can learn new skills by observing how people perform various activities. As humans, we have a remarkable ability to imitate other human’s behaviors and adapt to new situations. Endowing robots with these critical capabilities is a significant but very challenging problem considering the complexity and variation of human activities in highly dynamic environments. This research focuses on how robots can learn new skills by interpreting human activities, adapting the learned skills to new situations, and naturally interacting with humans. This dissertation begins with a discussion of challenges in each of these three problems. A new unified representation approach is introduced to enable robots to simultaneously interpret the high-level semantic meanings and generalize the low-level trajectories of a broad range of human activities. An adaptive framework based on feature space decomposition is then presented for robots to not only reproduce skills, but also autonomously and efficiently adjust the learned skills to new environments that are significantly different from demonstrations. To achieve natural Human Robot Interaction (HRI), this dissertation presents a Recurrent Neural Network based deep perceptual control approach, which is capable of integrating multi-modal perception sequences with actions for robots to interact with humans in long-term tasks. Overall, by combining the above approaches, an autonomous system is created for robots to acquire important skills that can be applied to human-centered applications. Finally, this dissertation concludes with a discussion of future directions that could accelerate the upcoming technological revolution of robot learning from human demonstration

Action recognition using deep learning

PhDIn this thesis we study deep learning architectures for the problem of human action recognition in image sequences, i.e. the problem of automatically recognizing what people are doing in a given video. As unlabeled video data is easily accessible these days, we first explore models that can learn meaningful representations of sequences without actually having to know what is happening in the sequences at hand. More specifically, we first explore the convolutional restricted Boltzmann machine (RBM) and show how a stack of convolutional RBMs can be used to learn and extract features from sequences in an unsupervised way. Using the classical Fisher vector pipeline to encode the extracted features we apply them on the task of action classification. We move on to feature extraction using larger, deep convolutional neural networks and propose a novel architecture which expresses the processing steps of the classical Fisher vector pipeline as network layers. By contrast to other methods where these steps are performed consecutively and the corresponding parameters are learned in an unsupervised manner, defining them as a single neural network allows us to refine the whole model discriminatively in an end to end fashion. We show that our method achieves significant improvements in comparison to the classical Fisher vector extraction chain and results in a comparable performance to other convolutional networks, while largely reducing the number of required trainable parameters. Finally, we explore how the proposed architecture can be modified into a hybrid network that combines the benefits of both unsupervised and supervised training methods, resulting in a model that learns a semi-supervised Fisher vector descriptor of the input data. We evaluate the proposed model at image classification and action recognition problems and show how the model's classification performance improves as the amount of unlabeled data increases during training