4 research outputs found
A New Low-Rank Tensor Model for Video Completion
In this paper, we propose a new low-rank tensor model based on the circulant
algebra, namely, twist tensor nuclear norm or t-TNN for short. The twist tensor
denotes a 3-way tensor representation to laterally store 2D data slices in
order. On one hand, t-TNN convexly relaxes the tensor multi-rank of the twist
tensor in the Fourier domain, which allows an efficient computation using FFT.
On the other, t-TNN is equal to the nuclear norm of block circulant
matricization of the twist tensor in the original domain, which extends the
traditional matrix nuclear norm in a block circulant way. We test the t-TNN
model on a video completion application that aims to fill missing values and
the experiment results validate its effectiveness, especially when dealing with
video recorded by a non-stationary panning camera. The block circulant
matricization of the twist tensor can be transformed into a circulant block
representation with nuclear norm invariance. This representation, after
transformation, exploits the horizontal translation relationship between the
frames in a video, and endows the t-TNN model with a more powerful ability to
reconstruct panning videos than the existing state-of-the-art low-rank models.Comment: 8 pages, 11 figures, 1 tabl
Non-convex Penalty for Tensor Completion and Robust PCA
In this paper, we propose a novel non-convex tensor rank surrogate function
and a novel non-convex sparsity measure for tensor. The basic idea is to
sidestep the bias of norm by introducing concavity. Furthermore, we
employ the proposed non-convex penalties in tensor recovery problems such as
tensor completion and tensor robust principal component analysis, which has
various real applications such as image inpainting and denoising. Due to the
concavity, the models are difficult to solve. To tackle this problem, we devise
majorization minimization algorithms, which optimize upper bounds of original
functions in each iteration, and every sub-problem is solved by alternating
direction multiplier method. Finally, experimental results on natural images
and hyperspectral images demonstrate the effectiveness and efficiency of the
proposed methods
Spatio-Temporal Tensor Sketching via Adaptive Sampling
Mining massive spatio-temporal data can help a variety of real-world
applications such as city capacity planning, event management, and social
network analysis. The tensor representation can be used to capture the
correlation between space and time and simultaneously exploit the latent
structure of the spatial and temporal patterns in an unsupervised fashion.
However, the increasing volume of spatio-temporal data has made it
prohibitively expensive to store and analyze using tensor factorization.
In this paper, we propose SkeTenSmooth, a novel tensor factorization
framework that uses adaptive sampling to compress the tensor in a temporally
streaming fashion and preserves the underlying global structure. SkeTenSmooth
adaptively samples incoming tensor slices according to the detected data
dynamics. Thus, the sketches are more representative and informative of the
tensor dynamic patterns. In addition, we propose a robust tensor factorization
method that can deal with the sketched tensor and recover the original
patterns. Experiments on the New York City Yellow Taxi data show that
SkeTenSmooth greatly reduces the memory cost and outperforms random sampling
and fixed rate sampling method in terms of retaining the underlying patterns
Learning tensors from partial binary measurements
In this paper we generalize the 1-bit matrix completion problem to higher
order tensors. We prove that when a bounded rank-, order- tensor
in can be estimated efficiently by only binary
measurements by regularizing its max-qnorm and M-norm as surrogates for its
rank. We prove that similar to the matrix case, i.e., when , the sample
complexity of recovering a low-rank tensor from 1-bit measurements of a subset
of its entries is the same as recovering it from unquantized measurements.
Moreover, we show the advantage of using 1-bit tensor completion over
matricization both theoretically and numerically. Specifically, we show how the
1-bit measurement model can be used for context-aware recommender systems.Comment: 26 page