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An excerpt from the 2024 Department Newsletter research article...

A tensor is a multi-way array of numbers. An order-1 tensor is just a vector u ∈ ℝⁿ. An order-2 tensor is a matrix M ∈ ℝ^{n1× n2} . An order-3 tensor is a 3-way array T ∈ ℝ^{n1× n2× n3} , and so on. [...]

Tensors have a bunch of applications in statistics and data science. For instance, certain datasets might naturally be represented as a 3-wayarray encoding 3-way interactions between 3 different variables. Another key example in statistics is the method of moments: given many samples of an n-dimensional random vector, it may be useful to compute the moments. The first moment is the mean (expected value), which is an n dimensional vector; the second moment is the n × n covariance matrix; the third moment is an n × n × n tensor; and so on. We are used to performing various primitive computations on matrices: eigenvalues and eigenvectors, singular value decomposition (SVD), low-rank approximation, and so on. For tensors of order 3 and above, the analogous operations tend to be much more difficult to compute, or even ill-defined.

For the full article, check out our 2024 Department Newsletter! This article starts on page 8.