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1. Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning

   Brand ; Kalev, Amir ; Li, Tongyang ; Lin, Cedric Yen-Yu ; Svore, Krysta M. ; Wu, Xiaodi ( July 2019 , Leibniz international proceedings in informatics)

   We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(logmore »m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest.« less
2. Minimum-gain Pole Placement with Sparse Static Feedback

   https://doi.org/10.1109/TAC.2020.3018615  

   Katewa, Vaibhav ; Pasqualetti, Fabio ( July 2020 , IEEE Transactions on Automatic Control)

   The minimum-gain eigenvalue assignment/pole placement problem (MGEAP) is a classical problem in LTI systems with static state feedback. In this paper, we study the MGEAP when the state feedback has arbitrary sparsity constraints. We formulate the sparse MGEAP problem as an equality-constrained optimization problem and present an analytical characterization of its locally optimal solution in terms of eigenvector matrices of the closed loop system. This result is used to provide a geometric interpretation of the solution of the non-sparse MGEAP, thereby providing additional insights for this classical problem. Further, we develop an iterative projected gradient descent algorithm to obtain local solutions for the sparse MGEAP using a parametrization based on the Sylvester equation. We present a heuristic algorithm to compute the projections, which also provides a novel method to solve the sparse EAP. Also, a relaxed version of the sparse MGEAP is presented and an algorithm is developed to obtain approximately sparse local solutions to the MGEAP. Finally, numerical studies are presented to compare the properties of the algorithms, which suggest that the proposed projec
3. Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form

   Bhojanapalli, Srinadh ; Boumal, Nicolas ; Jain, Prateek ; Netrapalli, Praneeth ( January 2018 , Proceedings of the 31st Conference On Learning Theory, PMLR)

   Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer–Monteiro factorization approach for solving SDPs. For a large class of SDPs, upon random perturbation of the cost matrix, with high probability, we show that all approximate second-order stationary points are approximate global optima for the penalty formulation of appropriately rank-constrained SDPs, as long as the number of constraints scales sub-quadratically with the desired rank. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion.
4. Sparse representation of vectors in lattices and semigroups

   https://doi.org/10.1007/s10107-021-01657-8  

   Aliev, Iskander ; Averkov, Gennadiy ; De Loera, Jesús A. ; Oertel, Timm ( May 2021 , Mathematical Programming)

   We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the$$\ell _0$$${\ell }_0$-norm of the vector. Our main results are new improved bounds on the minimal$$\ell _0$$${\ell }_0$-norm of solutions to systems$$A\varvec{x}=\varvec{b}$$$Ax=b$, where$$A\in \mathbb {Z}^{m\times n}$$$A\in Z^{m\times n}$,$${\varvec{b}}\in \mathbb {Z}^m$$$b\in Z^m$and$$\varvec{x}$$$x$is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with$$\ell _0$$${\ell }_0$-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over$$\mathbb {R}$$$R$, to other subdomains such as$$\mathbb {Z}$$$Z$. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.
5. A New Binary Programming Formulation and Social Choice Property for Kemeny Rank Aggregation

   https://doi.org/10.1287/deca.2021.0433  

   Yoo, Yeawon ; Escobedo, Adolfo R. ( September 2021 , Decision Analysis)

   Rank aggregation is widely used in group decision making and many other applications, where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives, contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem—whose ranking inputs may be complete and incomplete, with and without ties. Moreover, it leverages the equivalence of two ranking aggregation problems, namely, that of minimizing the Kemeny-Snell distance and of maximizing the Kendall-τ correlation, to compare the newly introduced binary programming formulation to a modified version of an existing integer programming formulation associated with the Kendall-τ distance. The new formulation has fewer variables and constraints, which leads to faster solution times. Moreover, we develop a new social choice property, the nonstrict extended Condorcet criterion, which can be regarded as a natural extension of the well-known Condorcet criterion and the Extended Condorcet criterion. Unlike its parent properties, themore »new property is adequate for handling complete rankings with ties. The property is leveraged to develop a structural decomposition algorithm, through which certain large instances of the NP-hard Kemeny rank aggregation problem can be solved exactly in a practical amount of time. To test the practical implications of the new formulation and social choice property, we work with instances constructed from a probabilistic distribution and with benchmark instances from PrefLib, a library of preference data.« less