Zeljka Stojanac

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Low-rank tensor recovery is an interesting subject from both
theoretical and application point of view. On one side, it is a
natural extension of the sparse vector and low-rank matrix recovery
problem. On the other side, estimating a low-rank tensor has
applications in many different areas such as machine learning, video
compression, and seismic data interpolation. In this talk, an
iterative approach to low rank tensor recovery is introduced. This
approach is a generalization of the iterative hard thresholding
algorithm for sparse vector and low-rank matrix recovery to tensor
scenario (tensor IHT or TIHT algorithm). Here, we consider the tensor
train decomposition (or TT-decomposition), also called the matrix
product states (or MPS). The a!
nalysis
of the algorithm is based on a
version of the restricted isometry property (tensor RIP or TRIP). We
show that subgaussian measurement ensembles satisfy TRIP with high
probability under an almost optimal condition on the number of
measurements. Additionally, we show that partial Fourier maps combined
with random sign flips of the tensor entries satisfy TRIP with high
probability. Under the assumption that the linear operator satisfies
TRIP and under an additional assumption on the thresholding operator,
we provide a linear convergence result for the TIHT algorithm. Similar
results also hold for HOSVD (Tucker) and Hierarchical Tucker
decomposition (HT-decomposition). Finally, we illustrate the
performance of iterative hard thresholding algorithm for tensor
recovery via numerical experiments where we consider recovery from
Gaussian measurement ensembles and Fourier measurements for third
order tensors.

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Universität Bonn
Seminarroom 0.02, ETP
Contact: not specified