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Scalable Low-Rank Semidefinite Programming for Certifiably Correct Machine Perception

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Algorithmic Foundations of Robotics XIV (WAFR 2020)

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Abstract

Semidefinite relaxation has recently emerged as a powerful technique for machine perception, in many cases enabling the recovery of certifiably globally optimal solutions of generally-intractable nonconvex estimation problems. However, the high computational cost of standard interior-point methods for semidefinite optimization prevents these algorithms from scaling effectively to the high-dimensional problems frequently encountered in machine perception tasks. To address this challenge, in this paper we present an efficient algorithm for solving the large-scale but low-rank semidefinite relaxations that underpin current certifiably correct machine perception methods. Our algorithm preserves the scalability of current state-of-the-art low-rank semidefinite optimizers that are custom-built for the geometry of specific machine perception problems, but generalizes to a broad class of convex programs over spectrahedral sets without the need for detailed manual analysis or design. The result is an easy-to-use, general-purpose computational tool capable of supporting the development and deployment of a broad class of novel certifiably correct machine perception methods.

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Notes

  1. 1.

    More precisely, it requires knowledge of an efficiently-computable retraction on that manifold [2].

  2. 2.

    In coordinates, \(([x]_{+})_i = \max (x_i, 0)\) for all \(i \in [n]\).

  3. 3.

    Here the LICQ is necessary to ensure that the Lagrange multipliers \((\lambda , \gamma )\) associated with Y are unique [44]. Without this condition, there could conceivably exist some other set of multipliers \((\lambda ', \gamma ') \in \mathbb {R}^{m_1} \times \mathbb {R}_{+}^{m_2}\) (that we do not have in hand) satisfying (8) and also (5), in which case X would be optimal for Problem 1. However, any multipliers satisfying (5) for \(X = YY^\mathsf {T}\) satisfy (8) a fortiori. Therefore, the uniqueness of \((\lambda , \gamma )\) implies that there cannot exist alternative multipliers \((\lambda ', \gamma ')\) satisfying (5), and therefore that X is not optimal (by Theorem 1).

  4. 4.

    More precisely, reference [14] studied (19) and (20) for the case of equality constraints (\(\varphi (X) = \Vert A(X) - b \Vert ^2\)), and derived the corresponding specialization of Theorem 5(a).

  5. 5.

    Note that the argument used to prove Theorem 4 does not directly apply to (19) and (20) because the inequality term \(\Vert \left[ \mathcal {B}(X) - u\right] _{+} \Vert ^2\) renders \(\varphi \) only \(C^1\). In the supplementary material, we show how to derive a \(C^\infty \) reformulation of (19) and (20).

  6. 6.

    We do not apply Algorithm 1 to (24) because (as described in Sect. 4.1) the data matrix \(\tilde{Q}\) is dense, and Algorithm 1 does not implement the specialized linear-algebraic subroutines developed in [33, 35, 36] that enable efficient operations with \(\tilde{Q}\).

  7. 7.

    Available at https://github.com/david-m-rosen/LowRankSDP.

  8. 8.

    Version 3.13.1, available at https://github.com/coin-or/Ipopt.

  9. 9.

    Available at https://spectralib.org/.

  10. 10.

    Version 6.2, available at https://www.pardiso-project.org/.

  11. 11.

    Available at https://github.com/david-m-rosen/SE-Sync.

  12. 12.

    This termination criterion only affects Alg1-T, since SE-Sync-S and SE-Sync-T employ Riemannian optimization methods that automatically enforce feasibility.

  13. 13.

    Note that the sum of the optimization and minimum-eigenvalue computation times is not equal to the total elapsed time for SE-Sync-S and SE-Sync-T because these algorithms also construct and cache certain sparse matrix factorizations [33, 36].

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Authors and Affiliations

  1. Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    David M. Rosen

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  1. David M. Rosen
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Correspondence to David M. Rosen .

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Editors and Affiliations

  1. Center for Ubiquitous Computing, University of Oulu, Oulu, Finland

    Steven M. LaValle

  2. Brendan Iribe Center for Computer Science and Engineering, University of Maryland, College Park, MD, USA

    Ming Lin

  3. Center for Ubiquitous Computing, University of Oulu, Oulu, Finland

    Timo Ojala

  4. Department of Computer Science and Engineering, Texas A&M University, College Station, TX, USA

    Dylan Shell

  5. Department of Computer Science, Rutgers University, Piscataway, NJ, USA

    Jingjin Yu

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Rosen, D.M. (2021). Scalable Low-Rank Semidefinite Programming for Certifiably Correct Machine Perception. In: LaValle, S.M., Lin, M., Ojala, T., Shell, D., Yu, J. (eds) Algorithmic Foundations of Robotics XIV. WAFR 2020. Springer Proceedings in Advanced Robotics, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-030-66723-8_33

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