A General Framework for Fast Stagewise Algorithms

Forward stagewise regression follows a very simple strategy for constructing a sequence of sparse regression estimates: it starts with all coefficients equal to zero, and iteratively updates the coefficient (by a small amount $\epsilon$) of the variable that achieves the maximal absolute inner product with the current residual. This procedure has an interesting connection to the lasso: under some conditions, it is known that the sequence of forward stagewise estimates exactly coincides with the lasso path, as the step size $\epsilon$ goes to zero. Furthermore, essentially the same equivalence holds outside of least squares regression, with the minimization of a differentiable convex loss function subject to an $\ell_1$ norm constraint (the stagewise algorithm now updates the coefficient corresponding to the maximal absolute component of the gradient). Even when they do not match their $\ell_1$-constrained analogues, stagewise estimates provide a useful approximation, and are computationally appealing. Their success in sparse modeling motivates the question: can a simple, effective strategy like forward stagewise be applied more broadly in other regularization settings, beyond the $\ell_1$ norm and sparsity? The current paper is an attempt to do just this. We present a general framework for stagewise estimation, which yields fast algorithms for problems such as group-structured learning, matrix completion, image denoising, and more.Comment: 56 pages, 15 figure

Multiframe Scene Flow with Piecewise Rigid Motion

We introduce a novel multiframe scene flow approach that jointly optimizes the consistency of the patch appearances and their local rigid motions from RGB-D image sequences. In contrast to the competing methods, we take advantage of an oversegmentation of the reference frame and robust optimization techniques. We formulate scene flow recovery as a global non-linear least squares problem which is iteratively solved by a damped Gauss-Newton approach. As a result, we obtain a qualitatively new level of accuracy in RGB-D based scene flow estimation which can potentially run in real-time. Our method can handle challenging cases with rigid, piecewise rigid, articulated and moderate non-rigid motion, and does not rely on prior knowledge about the types of motions and deformations. Extensive experiments on synthetic and real data show that our method outperforms state-of-the-art.Comment: International Conference on 3D Vision (3DV), Qingdao, China, October 201

Multiframe Scene Flow with Piecewise Rigid Motion

We introduce a novel multiframe scene flow approach that jointly optimizes the consistency of the patch appearances and their local rigid motions from RGB-D image sequences. In contrast to the competing methods, we take advantage of an oversegmentation of the reference frame and robust optimization techniques. We formulate scene flow recovery as a global non-linear least squares problem which is iteratively solved by a damped Gauss-Newton approach. As a result, we obtain a qualitatively new level of accuracy in RGB-D based scene flow estimation which can potentially run in real-time. Our method can handle challenging cases with rigid, piecewise rigid, articulated and moderate non-rigid motion, and does not rely on prior knowledge about the types of motions and deformations. Extensive experiments on synthetic and real data show that our method outperforms state-of-the-art.Comment: International Conference on 3D Vision (3DV), Qingdao, China, October 201

High-Dimensional Low-Rank Tensor Autoregressive Time Series Modeling

Modern technological advances have enabled an unprecedented amount of structured data with complex temporal dependence, urging the need for new methods to efficiently model and forecast high-dimensional tensor-valued time series. This paper provides the first practical tool to accomplish this task via autoregression (AR). By considering a low-rank Tucker decomposition for the transition tensor, the proposed tensor autoregression can flexibly capture the underlying low-dimensional tensor dynamics, providing both substantial dimension reduction and meaningful dynamic factor interpretation. For this model, we introduce both low-dimensional rank-constrained estimator and high-dimensional regularized estimators, and derive their asymptotic and non-asymptotic properties. In particular, by leveraging the special balanced structure of the AR transition tensor, a novel convex regularization approach, based on the sum of nuclear norms of square matricizations, is proposed to efficiently encourage low-rankness of the coefficient tensor. A truncation method is further introduced to consistently select the Tucker ranks. Simulation experiments and real data analysis demonstrate the advantages of the proposed approach over various competing ones.Comment: 61 pages, 6 figure