Lijun Ding

Lijun Ding, Liwei Jiang, Yudong Chen et al.

We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust $\ell_1$ loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank. On the other hand, we show that the RDPP condition holds under generic settings, such as Gaussian measurements under independent or adversarial sparse corruptions, where the result could be of independent interest. Both the exact recovery and the convergence rate of the proposed subgradient method are numerically verified in the overspecified regime. Moreover, our experiment further shows that our particular design of diminishing stepsize effectively prevents overfitting for robust recovery under overparameterized models, such as robust matrix sensing and learning robust deep image prior. This regularization effect is worth further investigation.

Chengrun Yang, Lijun Ding, Ziyang Wu et al.

Tensors are widely used to represent multiway arrays of data. The recovery of missing entries in a tensor has been extensively studied, generally under the assumption that entries are missing completely at random (MCAR). However, in most practical settings, observations are missing not at random (MNAR): the probability that a given entry is observed (also called the propensity) may depend on other entries in the tensor or even on the value of the missing entry. In this paper, we study the problem of completing a partially observed tensor with MNAR observations, without prior information about the propensities. To complete the tensor, we assume that both the original tensor and the tensor of propensities have low multilinear rank. The algorithm first estimates the propensities using a convex relaxation and then predicts missing values using a higher-order SVD approach, reweighting the observed tensor by the inverse propensities. We provide finite-sample error bounds on the resulting complete tensor. Numerical experiments demonstrate the effectiveness of our approach.

Lijun Ding, Yuqian Zhang, Yudong Chen

Existing results for low-rank matrix recovery largely focus on quadratic loss, which enjoys favorable properties such as restricted strong convexity/smoothness (RSC/RSM) and well conditioning over all low rank matrices. However, many interesting problems involve more general, non-quadratic losses, which do not satisfy such properties. For these problems, standard nonconvex approaches such as rank-constrained projected gradient descent (a.k.a. iterative hard thresholding) and Burer-Monteiro factorization could have poor empirical performance, and there is no satisfactory theory guaranteeing global and fast convergence for these algorithms. In this paper, we show that a critical component in provable low-rank recovery with non-quadratic loss is a regularity projection oracle. This oracle restricts iterates to low-rank matrices within an appropriate bounded set, over which the loss function is well behaved and satisfies a set of approximate RSC/RSM conditions. Accordingly, we analyze an (averaged) projected gradient method equipped with such an oracle, and prove that it converges globally and linearly. Our results apply to a wide range of non-quadratic low-rank estimation problems including one bit matrix sensing/completion, individualized rank aggregation, and more broadly generalized linear models with rank constraints.

Lijun Ding, Jicong Fan, Madeleine Udell

This paper proposes a new variant of Frank-Wolfe (FW), called $k$FW. Standard FW suffers from slow convergence: iterates often zig-zag as update directions oscillate around extreme points of the constraint set. The new variant, $k$FW, overcomes this problem by using two stronger subproblem oracles in each iteration. The first is a $k$ linear optimization oracle ($k$LOO) that computes the $k$ best update directions (rather than just one). The second is a $k$ direction search ($k$DS) that minimizes the objective over a constraint set represented by the $k$ best update directions and the previous iterate. When the problem solution admits a sparse representation, both oracles are easy to compute, and $k$FW converges quickly for smooth convex objectives and several interesting constraint sets: $k$FW achieves finite $\frac{4L_f^3D^4}{γδ^2}$ convergence on polytopes and group norm balls, and linear convergence on spectrahedra and nuclear norm balls. Numerical experiments validate the effectiveness of $k$FW and demonstrate an order-of-magnitude speedup over existing approaches.

Lijun Ding, Madeleine Udell

Low rank matrix recovery problems appear widely in statistics, combinatorics, and imaging. One celebrated method for solving these problems is to formulate and solve a semidefinite program (SDP). It is often known that the exact solution to the SDP with perfect data recovers the solution to the original low rank matrix recovery problem. It is more challenging to show that an approximate solution to the SDP formulated with noisy problem data acceptably solves the original problem; arguments are usually ad hoc for each problem setting, and can be complex. In this note, we identify a set of conditions that we call simplicity that limit the error due to noisy problem data or incomplete convergence. In this sense, simple SDPs are robust: simple SDPs can be (approximately) solved efficiently at scale; and the resulting approximate solutions, even with noisy data, can be trusted. Moreover, we show that simplicity holds generically, and also for many structured low rank matrix recovery problems, including the stochastic block model, $\mathbb{Z}_2$ synchronization, and matrix completion. Formally, we call an SDP simple if it has a surjective constraint map, admits a unique primal and dual solution pair, and satisfies strong duality and strict complementarity. However, simplicity is not a panacea: we show the Burer-Monteiro formulation of the SDP may have spurious second-order critical points, even for a simple SDP with a rank 1 solution.

Jicong Fan, Lijun Ding, Yudong Chen et al.

This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the matrix rank function. Our new factor group-sparse regularizers are motivated as a relaxation of the number of nonzero columns in a factorization of the matrix. These nonconvex regularizers are sharper than the nuclear norm; indeed, we show they are related to Schatten-$p$ norms with arbitrarily small $0 < p \leq 1$. Moreover, these factor group-sparse regularizers can be written in a factored form that enables efficient and effective nonconvex optimization; notably, the method does not use singular value decomposition. We provide generalization error bounds for low-rank matrix completion which show improved upper bounds for Schatten-$p$ norm reglarization as $p$ decreases. Compared to the max norm and the factored formulation of the nuclear norm, factor group-sparse regularizers are more efficient, accurate, and robust to the initial guess of rank. Experiments show promising performance of factor group-sparse regularization for low-rank matrix completion and robust principal component analysis.

Lijun Ding, Benjamin Grimmer

In this paper, we show that the bundle method can be applied to solve semidefinite programming problems with a low rank solution without ever constructing a full matrix. To accomplish this, we use recent results from randomly sketching matrix optimization problems and from the analysis of bundle methods. Under strong duality and strict complementarity of SDP, our algorithm produces primal and the dual sequences converging in feasibility at a rate of $\tilde{O}(1/ε)$ and in optimality at a rate of $\tilde{O}(1/ε^2)$. Moreover, our algorithm outputs a low rank representation of its approximate solution with distance to the optimal solution at most $O(\sqrtε)$ within $\tilde{O}(1/ε^2)$ iterations.

Lijun Ding, Alp Yurtsever, Volkan Cevher et al.

This paper develops a new storage-optimal algorithm that provably solves generic semidefinite programs (SDPs) in standard form. This method is particularly effective for weakly constrained SDPs. The key idea is to formulate an approximate complementarity principle: Given an approximate solution to the dual SDP, the primal SDP has an approximate solution whose range is contained in the eigenspace with small eigenvalues of the dual slack matrix. For weakly constrained SDPs, this eigenspace has very low dimension, so this observation significantly reduces the search space for the primal solution. This result suggests an algorithmic strategy that can be implemented with minimal storage: (1) Solve the dual SDP approximately; (2) compress the primal SDP to the eigenspace with small eigenvalues of the dual slack matrix; (3) solve the compressed primal SDP. The paper also provides numerical experiments showing that this approach is successful for a range of interesting large-scale SDPs.

Lijun Ding, Yudong Chen

In this paper, we introduce a powerful technique based on Leave-one-out analysis to the study of low-rank matrix completion problems. Using this technique, we develop a general approach for obtaining fine-grained, entrywise bounds for iterative stochastic procedures in the presence of probabilistic dependency. We demonstrate the power of this approach in analyzing two of the most important algorithms for matrix completion: (i) the non-convex approach based on Projected Gradient Descent (PGD) for a rank-constrained formulation, also known as the Singular Value Projection algorithm, and (ii) the convex relaxation approach based on nuclear norm minimization (NNM). Using this approach, we establish the first convergence guarantee for the original form of PGD without regularization or sample splitting}, and in particular shows that it converges linearly in the infinity norm. For NNM, we use this approach to study a fictitious iterative procedure that arises in the dual analysis. Our results show that \NNM recovers an $ d $-by-$ d $ rank-$ r $ matrix with $\mathcal{O}(μr \log(μr) d \log d )$ observed entries. This bound has optimal dependence on the matrix dimension and is independent of the condition number. To the best of our knowledge, this is the first sample complexity result for a tractable matrix completion algorithm that satisfies these two properties simultaneously.