Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.Comment: 232 page

Towards Dual-functional Radar-Communication Systems: Optimal Waveform Design

We focus on a dual-functional multi-input-multi-output (MIMO) radar-communication (RadCom) system, where a single transmitter communicates with downlink cellular users and detects radar targets simultaneously. Several design criteria are considered for minimizing the downlink multi-user interference. First, we consider both the omnidirectional and directional beampattern design problems, where the closed-form globally optimal solutions are obtained. Based on these waveforms, we further consider a weighted optimization to enable a flexible trade-off between radar and communications performance and introduce a low-complexity algorithm. The computational costs of the above three designs are shown to be similar to the conventional zero-forcing (ZF) precoding. Moreover, to address the more practical constant modulus waveform design problem, we propose a branch-and-bound algorithm that obtains a globally optimal solution and derive its worst-case complexity as a function of the maximum iteration number. Finally, we assess the effectiveness of the proposed waveform design approaches by numerical results.Comment: 13 pages, 10 figures. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Randomized Local Model Order Reduction

In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.Comment: 31 pages, 14 figures, 1 table, 1 algorith

Nonlinear functional regression by functional deep neural network with kernel embedding

With the rapid development of deep learning in various fields of science and technology, such as speech recognition, image classification, and natural language processing, recently it is also widely applied in the functional data analysis (FDA) with some empirical success. However, due to the infinite dimensional input, we need a powerful dimension reduction method for functional learning tasks, especially for the nonlinear functional regression. In this paper, based on the idea of smooth kernel integral transformation, we propose a functional deep neural network with an efficient and fully data-dependent dimension reduction method. The architecture of our functional net consists of a kernel embedding step: an integral transformation with a data-dependent smooth kernel; a projection step: a dimension reduction by projection with eigenfunction basis based on the embedding kernel; and finally an expressive deep ReLU neural network for the prediction. The utilization of smooth kernel embedding enables our functional net to be discretization invariant, efficient, and robust to noisy observations, capable of utilizing information in both input functions and responses data, and have a low requirement on the number of discrete points for an unimpaired generalization performance. We conduct theoretical analysis including approximation error and generalization error analysis, and numerical simulations to verify these advantages of our functional net

Robust equalization of multichannel acoustic systems

In most real-world acoustical scenarios, speech signals captured by distant microphones from a source are reverberated due to multipath propagation, and the reverberation may impair speech intelligibility. Speech dereverberation can be achieved by equalizing the channels from the source to microphones. Equalization systems can be computed using estimates of multichannel acoustic impulse responses. However, the estimates obtained from system identification always include errors; the fact that an equalization system is able to equalize the estimated multichannel acoustic system does not mean that it is able to equalize the true system. The objective of this thesis is to propose and investigate robust equalization methods for multichannel acoustic systems in the presence of system identification errors. Equalization systems can be computed using the multiple-input/output inverse theorem or multichannel least-squares method. However, equalization systems obtained from these methods are very sensitive to system identification errors. A study of the multichannel least-squares method with respect to two classes of characteristic channel zeros is conducted. Accordingly, a relaxed multichannel least- squares method is proposed. Channel shortening in connection with the multiple- input/output inverse theorem and the relaxed multichannel least-squares method is discussed. Two algorithms taking into account the system identification errors are developed. Firstly, an optimally-stopped weighted conjugate gradient algorithm is proposed. A conjugate gradient iterative method is employed to compute the equalization system. The iteration process is stopped optimally with respect to system identification errors. Secondly, a system-identification-error-robust equalization method exploring the use of error models is presented, which incorporates system identification error models in the weighted multichannel least-squares formulation