Ke Wei

Ke Wei, Jian-Feng Cai, Tony F. Chan et al.

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an $m\times n$ rank $r$ matrix from $p < mn$ number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant $R_{3r}$ of the sensing operator is less than $C_κ/\sqrt{r}$, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank $r$ matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

Ke Wei, Jian-Feng Cai, Tony F. Chan et al.

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank $r$ matrix are sampled independently and uniformly with replacement. We first prove that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided \begin{align*} m\geq C_κn^{1.5}r\log^{1.5}(n), \end{align*} where $C_κ$ is a numerical constant depending on the condition number of the underlying matrix. The sampling complexity has been further improved to \begin{align*} m\geq C_κnr^2\log^{2}(n) \end{align*} via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements.

Jiacai Liu, Wenye Li, Ke Wei

Policy mirror descent (PMD) is a general policy optimization framework in reinforcement learning, which can cover a wide range of typical policy optimization methods by specifying different mirror maps. Existing analysis of PMD requires exact or approximate evaluation (for example unbiased estimation via Monte Carlo simulation) of action values solely based on policy. In this paper, we consider policy mirror descent with temporal difference evaluation (TD-PMD). It is shown that, given the access to exact policy evaluations, the dimension-free $O(1/T)$ sublinear convergence still holds for TD-PMD with any constant step size and any initialization. In order to achieve this result, new monotonicity and shift invariance arguments have been developed. The dimension free $γ$-rate linear convergence of TD-PMD is also established provided the step size is selected adaptively. For the two common instances of TD-PMD (i.e., TD-PQA and TD-NPG), it is further shown that they enjoy the convergence in the policy domain. Additionally, we investigate TD-PMD in the inexact setting and give the sample complexity for it to achieve the last iterate $\varepsilon$-optimality under a generative model, which improves the last iterate sample complexity for PMD over the dependence on $1/(1-γ)$.

Zhengyang Geng, Meng-Hao Guo, Hongxu Chen et al.

As an essential ingredient of modern deep learning, attention mechanism, especially self-attention, plays a vital role in the global correlation discovery. However, is hand-crafted attention irreplaceable when modeling the global context? Our intriguing finding is that self-attention is not better than the matrix decomposition (MD) model developed 20 years ago regarding the performance and computational cost for encoding the long-distance dependencies. We model the global context issue as a low-rank recovery problem and show that its optimization algorithms can help design global information blocks. This paper then proposes a series of Hamburgers, in which we employ the optimization algorithms for solving MDs to factorize the input representations into sub-matrices and reconstruct a low-rank embedding. Hamburgers with different MDs can perform favorably against the popular global context module self-attention when carefully coping with gradients back-propagated through MDs. Comprehensive experiments are conducted in the vision tasks where it is crucial to learn the global context, including semantic segmentation and image generation, demonstrating significant improvements over self-attention and its variants.

Jian-Feng Cai, Ke Wei

A Riemannian gradient descent algorithm and a truncated variant are presented to solve systems of phaseless equations $|Ax|^2=y$. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-$1$ positive semidefinite matrices. Theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. Two key ingredients in the analysis are the restricted well conditioned property and the restricted weak correlation property of the associated truncated linear operator. Empirical evaluations show that our algorithms are competitive with other state-of-the-art first order nonconvex approaches with provable guarantees.

Ke Wei

We study the Kaczmarz methods for solving systems of quadratic equations, i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz methods for solving systems of linear equations by integrating a phase selection heuristic in each iteration and overall have the same per iteration computational complexity. Extensive empirical performance comparisons establish the computational advantages of the Kaczmarz methods over other state-of-the-art phase retrieval algorithms both in terms of the number of measurements needed for successful recovery and in terms of computation time. Preliminary convergence analysis is presented for the randomized Kaczmarz methods.