Finding Low-Rank Solutions via Non-Convex Matrix Factorization, Efficiently and Provably

Park, Dohyung; Kyrillidis, Anastasios; Caramanis, Constantine; Sanghavi, Sujay

doi:10.48550/arxiv.1606.03168

Cited by 16 publications

(29 citation statements)

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“…For clarity, in Figure 2, we summarize the efficiency of momentum acceleration, by showing the reconstruction error only for the following settings: maxiters = 1000, η = 10 −3 , reltol = 5 × 10 −4 , and measpc = 20%. In the plots, µ = 0 corresponds to the FGD algorithm in [74], µ corresponds to the value obtained through our theory, while we use µ ∈ 1 8 , 1 4 , 1 3 , 3 4 to study the acceleration effect. For µ , per our theory, we follow the rule for µ; see also the Methods section for more details.…”

Section: Mifgd On 6-and 8-qubit Real Quantum Datamentioning

confidence: 99%

“…Effectively the factorization of a the set of PSD matrices to a product of rectangular matrices results in a non-convex setting. This approach have been heavily studied recently, due to computational and space complexity advantages [43,19,98,99,91,73,75,84,12,13,74,30,40,54]. None of the works above consider the inclusion and analysis of momentum.…”

Section: Related Workmentioning

confidence: 99%

“…The low-rankness of ρ is enforced through the factorization of the density matrix into a outer product of such a rectangular matrix representation U ∈ C d×r with its Hermitian conjugate, where d r. By rewriting ρ = U U † , for U ∈ C d×r , both the PSD constraint (ρ 0) and the low-rankness constraint (rank(ρ) ≤ r) are directly satisfied, leading to the non-convex formulation (3). Working in the factored space was shown [54,74,73,91,98,99] to improve time and space complexities.…”

Section: Introductionmentioning

confidence: 99%

“…This algorithm has been studied in [12,99,91,75,30,40]. We will refer to the above iteration as the Factored Gradient Descent (FGD) algorithm, as in [74]. In what follows, we will study the MiFGD algorithm, a momentum-inspired factored gradient descent.…”

Section: Introductionmentioning

confidence: 99%

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Fast quantum state reconstruction via accelerated non-convex programming

Kim¹,

Kollias²,

Kalev³

et al. 2021

Preprint

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We propose a new quantum state reconstruction method that combines ideas from compressed sensing, non-convex optimization, and acceleration methods. The algorithm, called Momentum-Inspired Factored Gradient Descent (MiFGD), extends the applicability of quantum tomography for larger systems. Despite being a non-convex method, MiFGD converges provably to the true density matrix at a linear rate, in the absence of experimental and statistical noise, and under common assumptions. With this manuscript, we present the method, prove its convergence property and provide Frobenius norm bound guarantees with respect to the true density matrix. From a practical point of view, we benchmark the algorithm performance with respect to other existing methods, in both synthetic and real experiments performed on an IBM's quantum processing unit. We find that the proposed algorithm performs orders of magnitude faster than state of the art approaches, with the same or better accuracy. In both synthetic and real experiments, we observed accurate and robust reconstruction, despite experimental and statistical noise in the tomographic data. Finally, we provide a ready-to-use code for state tomography of multi-qubit systems.

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Section: Mifgd On 6-and 8-qubit Real Quantum Datamentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast quantum state reconstruction via accelerated non-convex programming

Kim¹,

Kollias²,

Kalev³

et al. 2021

Preprint

Self Cite

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“…This reformulation has been thoroughly studied for various low rank minimization problems [31,39,14,44,15,4,42,38,44,35]. It has several benefits: first, the rank constraint is directly enforced by the factorization itself XX T .…”

Section: Introductionmentioning

confidence: 99%

Quartic First-Order Methods for Low-Rank Minimization

Dragomir,

d'Aspremont,

Bolte

2019

Preprint

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We study a generalized nonconvex Burer-Monteiro formulation for low-rank minimization problems. We use recent results on non-Euclidean first order methods to provide efficient and scalable algorithms. Our approach uses geometries induced by quartic kernels on matrix spaces; for unconstrained cases we introduce a novel family of Gram kernels that considerably improves numerical performances. Numerical experiments for Euclidean distance matrix completion and symmetric nonnegative matrix factorization show that our algorithms scale well and reach state of the art performance when compared to specialized methods. * Last two authors are listed in alphabetical order.Preprint. Under review.

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