In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.
We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.
We introduce a change of perspective on tensor network states that is defined by the computational graph of the contraction of an amplitude. The resulting class of states, which we refer to as tensor network functions, inherit the conceptual advantages of tensor network states while removing computational restrictions arising from the need to converge approximate contractions. We use tensor network functions to compute strict variational estimates of the energy on loopy graphs, analyze their expressive power for ground states, show that we can capture aspects of volume law time evolution, and provide a mapping of general feed-forward neural nets onto efficient tensor network functions. Our work expands the realm of computable tensor networks to ones where accurate contraction methods are not available, and opens up new avenues to use tensor networks.
In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor (which we will see are all the same object in this context) has \emph{low rank} may be ill-justified. There are many natural instances where the object in question has high rank with respect to the classical notions of rank: matrix rank, tensor rank, multilinear rank --- the latter two being the most straightforward generalizations of the former. To remedy this, we show that one may vastly expand these classical notions of ranks: Given any undirected graph
G, there is a notion of
G-rank associated with
G, which provides us with as many different kinds of ranks as there are undirected graphs. In particular, the popular tensor network states in physics (e.g., \textsc{mps}, \textsc{ttns}, \textsc{peps}) may be regarded as functions of a specific
G-rank for various choices of
G. Among other things, we will see that a function, matrix, or tensor may have very high matrix, tensor, or multilinear rank and yet very low
G-rank for some
G. In fact the difference is in the orders of magnitudes and the gaps between
G-ranks and these classical ranks are arbitrarily large for some important objects in computer science, mathematics, and physics. Furthermore, we show that there is a
G such that almost every tensor has
G-rank exponentially lower than its rank or the dimension of its ambient space.
Tensor network, which originates from quantum physics, is emerging as an efficient tool for classical and quantum machine learning. Nevertheless, there still exists a considerable accuracy gap between tensor network and the sophisticated neural network models for classical machine learning. In this work, we combine the ideas of matrix product state (MPS), the simplest tensor network structure, and residual neural network and propose the residual matrix product state (ResMPS). The ResMPS can be treated as a network where its layers map the "hidden" features to the outputs (e.g., classifications), and the variational parameters of the layers are the functions of the features of the samples (e.g., pixels of images). This is different from neural network, where the layers map feed-forwardly the features to the output. The ResMPS can equip with the non-linear activations and dropout layers, and outperforms the state-of-the-art tensor network models in terms of efficiency, stability, and expression power. Besides, ResMPS is interpretable from the perspective of polynomial expansion, where the factorization and exponential machines naturally emerge. Our work contributes to connecting and hybridizing neural and tensor networks, which is crucial to further enhance our understanding of the working mechanisms and improve the performance of both models.
Tensor network methods have been a key ingredient of advances in condensed matter physics and have recently sparked interest in the machine learning community for their ability to compactly represent very high-dimensional objects. Tensor network methods can for example be used to efficiently learn linear models in exponentially large feature spaces [Stoudenmire and Schwab, 2016]. In this work, we derive upper and lower bounds on the VC dimension and pseudo-dimension of a large class of tensor network models for classification, regression and completion. Our upper bounds hold for linear models parameterized by arbitrary tensor network structures, and we derive lower bounds for common tensor decomposition models~(CP, Tensor Train, Tensor Ring and Tucker) showing the tightness of our general upper bound. These results are used to derive a generalization bound which can be applied to classification with low rank matrices as well as linear classifiers based on any of the commonly used tensor decomposition models. As a corollary of our results, we obtain a bound on the VC dimension of the matrix product state classifier introduced in [Stoudenmire and Schwab, 2016] as a function of the so-called bond dimension~(i.e. tensor train rank), which answers an open problem listed by Cirac, Garre-Rubio and Pérez-García in [Cirac et al., 2019].