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1. Engineering Practical Rank-Code-Based Cryptographic Schemes on Embedded Hardware. A Case Study on ROLLO

   https://doi.org/10.1109/TC.2022.3225080  

   Hu, Jingwei; Wang, Wen; Gaj, Kris; Wang, Liping; Wang, Huaxiong (July 2023, IEEE Transactions on Computers)

   In this paper, we investigate the practical performance of rank-code based cryptography on FPGA platforms by presenting a case study on the quantum-safe KEM scheme based on LRPC codes called ROLLO, which was among NIST post-quantum cryptography standardization round-2 candidates. Specifically, we present an FPGA implementation of the encapsulation and decapsulation operations of the ROLLO KEM scheme with some variations to the original specification. The design is fully parameterized, using code-generation scripts to support a wide range of parameter choices for security levels specified in ROLLO. At the core of the ROLLO hardware, we presented a generic approach for hardware-based Gaussian elimination, which can process both non-singular and singular matrices. Previous works on hardware-based Gaussian elimination can only process non-singular ones. However, a plethora of cryptosystems, for instance, quantum-safe key encapsulation mechanisms based on rank-metric codes, ROLLO and RQC, which are among NIST post-quantum cryptography standardization round-2 candidates, require performing Gaussian elimination for random matrices regardless of the singularity. To the best of our knowledge, this work is the first hardware implementation for rank-code-based cryptographic schemes. The experimental results suggest rank-code-based schemes can be highly efficient. 

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2. Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning

   https://doi.org/10.1145/3549524  

   Chia, Nai-Hui; Gilyén, András Pal; Li, Tongyang; Lin, Han-Hsuan; Tang, Ewin; Wang, Chunhao (October 2022, Journal of the ACM)

   We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ²-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive. 

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3. A Unified Framework for Nonconvex Low-Rank plus Sparse Matrix Recovery

   Zhang, Xiao; Wang, Lingxiao Wang; Gu, Quanquan (April 2018, International Conference on Artificial Intelligence and Statistics)

   We propose a unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness conditions. Based on projected gradient descent and the double thresholding operator, our proposed generic algorithm is guaranteed to converge to the unknown low-rank and sparse matrices at a locally linear rate, while matching the best-known robustness guarantee (i.e., tolerance for sparsity). At the core of our theory is a novel structural Lipschitz gradient condition for low-rank plus sparse matrices, which is essential for proving the linear convergence rate of our algorithm, and we believe is of independent interest to prove fast rates for general superposition-structured models. We illustrate the application of our framework through two concrete examples: robust matrix sensing and robust PCA. Empirical experiments corroborate our theory. 

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4. 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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5. Efficient and scalable wave function compression using corner hierarchical matrices

   https://doi.org/10.1063/5.0231409  

   Berard, Kenneth O; Gao, Hongji; Teplukhin, Alexander; Jiao, Xiangmin; Levine, Benjamin G (November 2024, The Journal of Chemical Physics)

   The exponential scaling of complete active space and full configuration interaction (CI) calculations limits the ability of quantum chemists to simulate the electronic structures of strongly correlated systems. Herein, we present corner hierarchically approximated CI (CHACI), an approach to wave function compression based on corner hierarchical matrices (CH-matrices)—a new variant of hierarchical matrices based on block-wise low-rank decomposition. By application to dodecacene, a strongly correlated molecule, we demonstrate that CH matrix compression provides superior compression compared to truncated global singular value decomposition. The compression ratio is shown to improve with increasing active space size. By comparison of several alternative schemes, we demonstrate that superior compression is achieved by (a) using a blocking approach that emphasizes the upper-left corner of the CI vector, (b) sorting the CI vector prior to compression, and (c) optimizing the rank of each block to maximize information density. 

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