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1. Robust Matrix Sensing in the Semi-Random Model

   Gao, Xing ; Cheng, Yu ( December 2023 , Proceedings of the 37th Conference on Neural Information Processing Systems)

   Low-rank matrix recovery is a fundamental problem in machine learning with numerous applications. In practice, the problem can be solved by convex optimization namely nuclear norm minimization, or by non-convex optimization as it is well-known that for low-rank matrix problems like matrix sensing and matrix completion, all local optima of the natural non-convex objectives are also globally optimal under certain ideal assumptions. In this paper, we study new approaches for matrix sensing in a semi-random model where an adversary can add any number of arbitrary sensing matrices. More precisely, the problem is to recover a low-rank matrix $X^\star$ from linear measurements $b_i = \langle A_i, X^\star \rangle$, where an unknown subset of the sensing matrices satisfies the Restricted Isometry Property (RIP) and the rest of the $A_i$'s are chosen adversarially. It is known that in the semi-random model, existing non-convex objectives can have bad local optima. To fix this, we present a descent-style algorithm that provably recovers the ground-truth matrix $X^\star$. For the closely-related problem of semi-random matrix completion, prior work [CG18] showed that all bad local optima can be eliminated by reweighting the input data. However, the analogous approach for matrix sensing requires reweighting a set of matrices to satisfy RIP, which is a condition that is NP-hard to check. Instead, we build on the framework proposed in [KLL$^+$23] for semi-random sparse linear regression, where the algorithm in each iteration reweights the input based on the current solution, and then takes a weighted gradient step that is guaranteed to work well locally. Our analysis crucially exploits the connection between sparsity in vector problems and low-rankness in matrix problems, which may have other applications in obtaining robust algorithms for sparse and low-rank problems. 

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2. Online Network Topology Inference with Partial Connectivity Information

   https://doi.org/10.1109/CAMSAP45676.2019.9022677  

   Shafipour, Rasoul ; Mateos, Gonzalo ( December 2019 , 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP))

   We develop algorithms for online topology inference from streaming nodal observations and partial connectivity information; i.e., a priori knowledge on the presence or absence of a few edges may be available as in the link prediction problem. The observations are modeled as stationary graph signals generated by local diffusion dynamics on the unknown network. Said stationarity assumption implies the simultaneous diagonalization of the observations' covariance matrix and the so-called graph shift operator (GSO), here the adjacency matrix of the sought graph. When the GSO eigenvectors are perfectly obtained from the ensemble covariance, we examine the structure of the feasible set of adjacency matrices and its dependency on the prior connectivity information available. In practice one can only form an empirical estimate of the covariance matrix, so we develop an alternating algorithm to find a sparse GSO given its imperfectly estimated eigenvectors. Upon sensing new diffused observations in the streaming setting, we efficiently update eigenvectors and perform only one (or a few) online iteration(s) of the proposed algorithm until a new datum is observed. Numerical tests showcase the effectiveness of the novel batch and online algorithms in recovering real-world graphs. 

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3. Hardware accelerator thread for unstructured sparse data processing

   Pranathi Vasireddy, Krishna Kavi ( October 2022 , International Conference on Computer Aided Design)

   Sparse matrix-dense vector (SpMV) multiplication is inherent in most scientific, neural networks and machine learning algorithms. To efficiently exploit sparsity of data in the SpMV computations, several compressed data representations have been used. However, the compressed data representations of sparse date can result in overheads for locating nonzero values, requiring indirect memory accesses and increased instruction count and memory access delays. We call these translations of compressed representations as metadata processing. We propose a memory-side accelerator for metadata (or indexing) computations and supplying only the required nonzero values to the processor, additionally permitting an overlap of indexing with core computations on nonzero elements. In this contribution, we target our accelerator for low-end microcontrollers with very limited memory and processing capabilities. In this paper we will explore two dedicated ASIC designs of the proposed accelerator that handles the indexed memory accesses for compressed sparse row (CSR) format working alongside a simple RISC-like programmable core. One version of the the accelerator supplies only vector values corresponding to nonzero matrix values and the second version supplies both nonzero matrix and matching vector values for SpMV computations. Our experiments show speedups ranging between 1.3 and 2.1 times for SpMV for different levels of sparsities. Our accelerator also results in energy savings ranging between 15.8% and 52.7% over different matrix sizes, when compared to the baseline system with primary RISC-V core performing all computations. We use smaller synthetic matrices with different sparsities and larger real-world matrices with higher sparsities (below 1% non-zeros) in our experimental evaluations. 

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4. High-Rank Matrix Completion with Side Information

   Wang, Y. ; Elhamifar, E. ( January 2018 , AAAI Conference on Artificial Intelligence)

   We address the problem of high-rank matrix completion with side information. In contrast to existing work dealing with side information, which assume that the data matrix is low-rank, we consider the more general scenario where the columns of the data matrix are drawn from a union of low-dimensional subspaces, which can lead to a high rank matrix. Our goal is to complete the matrix while taking advantage of the side information. To do so, we use the self-expressive property of the data, searching for a sparse representation of each column of matrix as a combination of a few other columns. More specifically, we propose a factorization of the data matrix as the product of side information matrices with an unknown interaction matrix, under which each column of the data matrix can be reconstructed using a sparse combination of other columns. As our proposed optimization, searching for missing entries and sparse coefficients, is non-convex and NP-hard, we propose a lifting framework, where we couple sparse coefficients and missing values and define an equivalent optimization that is amenable to convex relaxation. We also propose a fast implementation of our convex framework using a Linearized Alternating Direction Method. By extensive experiments on both synthetic and real data, and, in particular, by studying the problem of multi-label learning, we demonstrate that our method outperforms existing techniques in both low-rank and high-rank data regimes. 

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5. High-Rank Matrix Completion with Side Information

   Wang, Y. ; Elhamifar, E. ( January 2018 , AAAI Conference on Artificial Intelligence)

   We address the problem of high-rank matrix completion with side information. In contrast to existing work dealing with side information, which assume that the data matrix is low-rank, we consider the more general scenario where the columns of the data matrix are drawn from a union of low-dimensional subspaces, which can lead to a high rank matrix. Our goal is to complete the matrix while taking advantage of the side information. To do so, we use the self-expressive property of the data, searching for a sparse representation of each column of matrix as a combination of a few other columns. More specifically, we propose a factorization of the data matrix as the product of side information matrices with an unknown interaction matrix, under which each column of the data matrix can be reconstructed using a sparse combination of other columns. As our proposed optimization, searching for missing entries and sparse coefficients, is non-convex and NP-hard, we propose a lifting framework, where we couple sparse coefficients and missing values and define an equivalent optimization that is amenable to convex relaxation. We also propose a fast implementation of our convex framework using a Linearized Alternating Direction Method. By extensive experiments on both synthetic and real data, and, in particular, by studying the problem of multi-label learning, we demonstrate that our method outperforms existing techniques in both low-rank and high-rank data regimes 

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