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#Tensor Network Presentation

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- TENSOR NETWORK
- Tensor Tensor: natural presentation of multi-dimensional array A D-order Tensor is denoted as
- Tensor Network Tensor Network (TNs): ● Sparse data structures engineered for the efficient representation and manipulation of very high-dimensional Tensor. ● Graphical representation with Tensor Network Diagram. Scalar I1 I2 I3 I J I I I J I1 I2 I3 Vector Matrix Cube Rank-0 Tensor Rank-1 Tensor Rank-2 Tensor Rank-3 Tensor Array form Graphical Diagram Form
- Tensor Network Tensor Operations A x I J I b=Ax A B C=AB I J K I K A B I J K L M P C L M P I J Matrix-Vector Multiplication Matrix-Matrix Multiplication Matrix Trace Tensor-Tensor Contraction
- Tensor Network Tensor Decomposition I1 I2 I3 I4 I5 I6 1 1 1 1 1 1 I1 I2 I3 I4 I5 I6 r r r r r r I1 I2 I3 I4 I5 I6 r r r r r I1 I2 I3 I4 I5 I6 r r r r r r CP Decomposition Tucker Decomposition Tensor Train Decomposition Tensor Ring Decomposition
- Types of Tensor Network Matrix Product States / Tensor Train Decomposition Matrix Product States (MPS) is a factorization of a tensor with N indices into a chain-like product of three-index tensors. MPS MPS with periodic boundary conditions
- Types of Tensor Network Matrix Product Operator A matrix product operator (MPO) is a tensor network where each tensor has two external, uncontracted indices as well as two internal indices contracted with neighboring tensors in a chain-like fashion. MPO
- Types of Tensor Network Tree Tensor Network / Hierarchical Tucker Decomposition Tree Tensor Network is class of Tensor Network presented by general tree-like structures.
- Types of Tensor Network Projected Entangled Pair States (PEPS) The PEPS tensor network generalizes the Matrix Product State / Tensor Train Tensor Network from a one-dimensional network to a network on an arbitrary graph. The tensor diagram for a PEPS on a finite square lattice is:
- Types of Tensor Network Multi-scale Entanglement Renormalization Ansatz (MERA) Compared with the PEPS formats, the main advantage of the MERA formats is that the order and size of each core tensor in the internal tensor network structure is often much smaller, which dramatically reduces the number of free parameters and provides more efficient distributed storage of huge-scale data tensors.
- Applications of Tensor Network Big domains: ● Representing Continuous Functions ● Machine Learning ● Probability and Statistic ● Quantum Physics In Machine Learning: ● Quantum Computers ● Supervised/ Unsupervised/ Reinforcement Learning ● Expressivity & priors of TN based models ● Generative models ● Compressing weights of neural nets ● Optimization Methods ● Feature extraction & tensor completion ● …
- Resource https://tensornetwork.org/ [1] Roberts, Chase, et al. "Tensornetwork: A library for physics and machine learning." arXiv preprint arXiv:1905.01330 (2019). [2] Biamonte, Jacob. "Lectures on quantum tensor networks." arXiv preprint arXiv:1912.10049 (2019). [3] Biamonte, Jacob, and Ville Bergholm. "Tensor networks in a nutshell." arXiv preprint arXiv:1708.00006 (2017). [4] Orús, Román. "A practical introduction to tensor networks: Matrix product states and projected entangled pair states." Annals of physics 349 (2014): 117-158. [5] Bridgeman, Jacob C., and Christopher T. Chubb. "Hand-waving and interpretive dance: an introductory course on tensor networks." Journal of physics A: Mathematical and theoretical 50.22 (2017): 223001. [6] Cichocki, Andrzej, et al. "Low-rank tensor networks for dimensionality reduction and large-scale optimization problems: Perspectives and challenges part 1." arXiv preprint arXiv:1609.00893 (2016).

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