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PARAFAC and ALS for Low-Rank Decomposition in Multi-Way Tensor Data Analysis

This document outlines the methodology employed in DAFNE WP1.7, focusing on PARAFAC and ALS techniques for low-rank decomposition of multi-way data tensors. Data configurations include three-dimensional tensors representing various applications such as gastronomy and metabolomics, involving significant datasets reaching approximately 1 TB. The study covers crucial aspects such as data identifiability, resampling methods, and handling incomplete data with up to 90% missing values, applying rigorous iterative fitting processes to enhance statistical reliability.

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PARAFAC and ALS for Low-Rank Decomposition in Multi-Way Tensor Data Analysis

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  1. Maxeler @ Pupin Institute Goran Dimić goran.dimic@pupin.rs

  2. DAFNE WP1.7: PARAFAC, ALS • 3-way data tensors: low-rank decomposition • In: X: I x J x K • Out: A: I x F, B: J x F, C: K x F • Identifiability: Unique under mild conditions • Repeat: 1. A(X, B, C); 2. B(X, C, A); 3. C(X, A, B);

  3. DAFNE WP1.7: Data, Applications • Gastronomy: 42.000 x 350 x 24.000 (UCPH) • Metabolomics: 3.600 x 9.500 x 1.600 (UCPH) • Data size: ~1 TB • Resampling: 1.000x (for data statistics) ~1 PB • Fit each sampling: 1.000 – 100.000 iterations • Equivalent data size: ~1+ EB • Incomplete data: fit if up to 90% missing • Applications: (Hyper)graphs, data mining, SP…

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