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Register a new userOptimal Estimation of Low Rank Density Matrices
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Added by
Dong Xia
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters. Such bounds are established for several statistically relevant distances, including quant
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Let ${mathcal S}_m$ be the set of all $mtimes m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $rhoin {mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,dots, X_nin {mathbb H}_m$ (${mathbb H}_m$ being the space of $mtimes m$ Hermitian matrices) for a quantum system identically prepared $n$ times in state $rho.$ Outcomes $Y_1,dots, Y_n$ of such measurements could be described by a trace regression model in which ${mathbb E}_{rho}(Y_j|X_j)={rm tr}(rho X_j), j=1,dots, n.$ The design variables $X_1,dots, X_n$ are often sampled at random from the uniform distribution in an orthonormal basis ${E_1,dots, E_{m^2}}$ of ${mathbb H}_m$ (such as Pauli basis). The goal is to estimate the unknown density matrix $rho$ based on the data $(X_1,Y_1), dots, (X_n,Y_n).$ Let $$ hat Z:=frac{m^2}{n}sum_{j=1}^n Y_j X_j $$ and let $check rho$ be the projection of $hat Z$ onto the convex set ${mathcal S}_m$ of density matrices. It is shown that for estimator $check rho$ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten $p$-norm distances, $pin [1,infty]$ and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator $check rho$ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one projections of an approximately low-rank PSD matrix, we characterize the radius of the set of PSD matrices that satisfy the measurements. This result yields a sampling rate to guarantee singleton solution sets when the true matrix is exactly low-rank, such that the choice of the objective function or the algorithm to be used is inconsequential in its recovery. We discuss applications of this contribution and compare it to recent literature regarding implicit regularization for similar problems. We demonstrate practical implications of this result by applying conic projection methods for PSD matrix recovery without incorporating low-rank regularization.
A distance matrix $A in mathbb R^{n times m}$ represents all pairwise distances, $A_{ij}=mathrm{d}(x_i,y_j)$, between two point sets $x_1,...,x_n$ and $y_1,...,y_m$ in an arbitrary metric space $(mathcal Z, mathrm{d})$. Such matrices arise in various computational contexts such as learning image manifolds, handwriting recognition, and multi-dimensional unfolding. In this work we study algorithms for low-rank approximation of distance matrices. Recent work by Bakshi and Woodruff (NeurIPS 2018) showed it is possible to compute a rank-$k$ approximation of a distance matrix in time $O((n+m)^{1+gamma}) cdot mathrm{poly}(k,1/epsilon)$, where $epsilon>0$ is an error parameter and $gamma>0$ is an arbitrarily small constant. Notably, their bound is sublinear in the matrix size, which is unachievable for general matrices. We present an algorithm that is both simpler and more efficient. It reads only $O((n+m) k/epsilon)$ entries of the input matrix, and has a running time of $O(n+m) cdot mathrm{poly}(k,1/epsilon)$. We complement the sample complexity of our algorithm with a matching lower bound on the number of entries that must be read by any algorithm. We provide experimental results to validate the approximation quality and running time of our algorithm.
In this paper, we propose three approaches for the estimation of the Tucker decomposition of multi-way arrays (tensors) from partial observations. All approaches are formulated as convex minimization problems. Therefore, the minimum is guaranteed to be unique. The proposed approaches can automatically estimate the number of factors (rank) through the optimization. Thus, there is no need to specify the rank beforehand. The key technique we employ is the trace norm regularization, which is a popular approach for the estimation of low-rank matrices. In addition, we propose a simple heuristic to improve the interpretability of the obtained factorization. The advantages and disadvantages of three proposed approaches are demonstrated through numerical experiments on both synthetic and real world datasets. We show that the proposed convex optimization based approaches are more accurate in predictive performance, faster, and more reliable in recovering a known multilinear structure than conventional approaches.
Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform simple spectral approaches. Recent work characterizes the asymptotic accuracy of Bayes-optimal estimators in the high-dimensional limit. In this paper we present a practical algorithm that can achieve Bayes-optimal accuracy above the spectral threshold. A bold conjecture from statistical physics posits that no polynomial-time algorithm achieves optimal error below the same threshold (unless the best estimator is trivial). Our approach uses Approximate Message Passing (AMP) in conjunction with a spectral initialization. AMP algorithms have proved successful in a variety of statistical estimation tasks, and are amenable to exact asymptotic analysis via state evolution. Unfortunately, state evolution is uninformative when the algorithm is initialized near an unstable fixed point, as often happens in low-rank matrix estimation. We develop a new analysis of AMP that allows for spectral initializations. Our main theorem is general and applies beyond matrix estimation. However, we use it to derive detailed predictions for the problem of estimating a rank-one matrix in noise. Special cases of this problem are closely related---via universality arguments---to the network community detection problem for two asymmetric communities. For general rank-one models, we show that AMP can be used to construct confidence intervals and control false discovery rate. We provide illustrations of the general methodology by considering the cases of sparse low-rank matrices and of block-constant low-rank matrices with symmetric blocks (we refer to the latter as to the `Gaussian Block Model).
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