Gaussian Measurements Canyi Lu 1 , Jiashi Feng 2 , Zhouchen Lin 3 , - PowerPoint PPT Presentation

Exact Low Tubal Rank Tensor Recovery ry from Gaussian Measurements Canyi Lu 1 , Jiashi Feng 2 , Zhouchen Lin 3 , Shuicheng Yan 2 1 Carnegie Mellon University 2 National University of Singapore 3 Peking University

Low dimensional structures in visual data Learning by using the underlying low dimensional structure of data is important.

Compressive Sensing • Compressive sensing: learning by using sparse vector structure • Face recognition (J. Wright, et al., TPAMI, 2009)

Low-rank Matrix Recovery • Low-rank matrix: sparse singular values • Low-rank structure is common in visual data • Low-rank models, e.g., robust PCA, and matrix completion, have many applications • Background modeling • Removing shadows from face images • Image alignment • Many others… E. J. Cand` es, X. D. Li, Y. Ma, and J. Wright. Robust principal component analysis? Journal of the ACM, 2011

Multi-dimensional Data: Tensor

Structured Sparsity Sparse vector Low rank matrix Low rank tensor This work

Low-rank Tensor Learning Is Challenging • The tensor rank and tensor nuclear norm are not well defined • Tensor CP-rank and its convex envelop are NP-hard to compute • Tucker rank and Sum of Nuclear Norm (SNN) • SNN is a loose convex surrogate of Tucker rank • Recently, we propose a new tensor nuclear norm induced by tensor- tensor product for low tubal rank recovery Canyi Lu,et al.. Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization. CVPR. 2016.

Notations • Block circulant matrix of frontal slices • Two operators

Tensor-Tensor Product • Tensor-tensor product is a natural extension of matrix-matrix product. Misha E Kilmer and Carla D Martin. Factorization strategies for third-order tensors. Linear Algebra and its Applications, 2011

Tensor-SVD Canyi Lu,et al.. Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization. CVPR. 2016.

Problem I: Low-rank Tensor Recovery from Gaussian Measurements • Given a linear map and the observations for with tubal rank • Goal: to recover the low-rank tensor from the observations • Method: recovery by convex optimization • Question: what is the number of measurements required for exact recovery, i.e., ?

Main Result: Low-rank Tensor Recovery from Gaussian Measurements • For Gaussian measurements, the recovery is exact by convex optimization. • The required number of measurements is which is order optimal.

Problem II: Low-rank Tensor Completion • Given an incomplete tensor with tubal rank • Goal: to recover the low-rank tensor from partial observations • Method: recovery by convex optimization • Question: any exact recovery guarantee by convex optimization?

Main Result: Low-rank Tensor Completion • Exact recovery when the sampling complexity is of the order

Experiment: recovery from Gaussian measurements Exact recovery

Exact Experiment: low-rank tensor completion recovery

Experiment: tensor completion for image recovery

Experiment: tensor completion for video recovery

Conclusions • Tensor nuclear norm is a recently proposed convex surrogate for the pursuit of tensor tubal rank induced by the tensor-tensor product • Theoretical guarantee for low tubal rank tensor recovery from Gaussian measurements • Theoretical guarantee for low tubal rank tensor completion

Gaussian Measurements Canyi Lu 1 , Jiashi Feng 2 , Zhouchen Lin 3 , - PowerPoint PPT Presentation

Exact Low Tubal Rank Tensor Recovery ry from Gaussian Measurements Canyi Lu 1 , Jiashi Feng 2 , Zhouchen Lin 3 , Shuicheng Yan 2 1 Carnegie Mellon University 2 National University of Singapore 3 Peking University Low dimensional structures in

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