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Convex Multilinear Estimation with Tensors
Tensors are a powerful way to represent and analyze the most diverse type of data with applications in image and video recognition, EEG and fMRI, text analysis and recommendation systems, to name a few. In this presentation we go through four parts: A) We begin with a bird-eye view over this domain: what are tensors? why are they useful? what are the established approaches in tensor-based data analysis? B) As it turns out tensor based techniques are mostly based on generalization of the singular value decomposition. In this work we take a different perspective and rely on convex optimization. We study a broad class of non-smooth convex optimization problems for tensors. A penalty based on nuclear norms is used to enforce solutions with small (multilinear) ranks. A simple yet effective algorithm, termed Convex MultiLinear Estimation (CMLE), is proposed. C) We show how this algorithm can be specialized to accomplish different data-driven modeling tasks. Extending the existing taxonomy of learning to the case where input (and possibly output) patterns are represented as tensors, we can called these problems unsupervised or supervised. This generalization is instrumental to deal with important aspects - often overlooked in the tensor literature - such as the choice of loss functions, model selection, regularization and out-of-sample extensions. D) We present concrete examples ranging from image and video completion to low rank denoising and classification. A particular attention is devoted to present a case study on EEG data. Several epileptic seizure detection systems apply traditional machine learning techniques to differentiate between ictal and non-ictal EEG segments. They generally work based upon single channels and lead to ignore the spatial distribution of the ictal pattern. We show how a tensor based learning approach can overcome these limitations.