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# Tensor Factorization

November 26, 2023
written by Rida Mirza

Tensor factorization refers to methods for decomposing multi-dimensional arrays, known as tensors, into simpler components. Tensors can model complex multidimensional datasets found in areas like neuroscience, computer vision, natural language processing, and more.

## Key Methods for Tensor Factorization

• CANDECOMP/PARAFAC (CP) Decomposition: Factorizes a tensor into a sum of rank-one tensor components. Useful for identifying latent factors within the tensor.
• Tucker Decomposition: Decomposes a tensor into a core tensor multiplied by factor matrices along each mode. Captures interactions between modalities as well as latent factors.
• Tensor Train Decomposition: Represents a tensor by a sequence of third-order tensors that approximate the original tensor. Efficient for working with high-dimensional data.

### Example of CANDECOMP/PARAFAC (CP) Decomposition

Given a 100x100x100 tensor X, CP decomposition factorizes it as:

X = âˆ‘ A B C

Where A, B and C are 100×5 factor matrices. This approximates X as a sum of 5 rank-one tensors, revealing 5 latent factors that span the modes of X. The individual columns of A, B, C represent the factor loadings.

### Example of Tucker Decomposition

Given the same 100x100x100 tensor X, Tucker decomposition expresses it as:

X = G x1 A x2 B x3 C

Where G is a 5x5x5 core tensor, and A, B and C are 100×5 factor matrices like before. This models interactions between the latent factors via G, while still identifying interpretable factors.

## FAQS

### What types of data are suited for tensor factorization?

Multi-modal, multi-dimensional datasets like MRI scans, video data and multi-language corpora.

### What are the main advantages of tensor methods over matrix methods?

Ability to model multi-way structure and interactions, greater expressiveness.

### How can the results of tensor factorization be interpreted?

The factors often relate to latent semantic or explanatory factors within the data

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