We present two open-source implementations of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) algorithm to find a few eigenvalues and eigenvectors of large, possibly sparse matrices. We then test LOBPCG for various quantum chemistry problems, encompassing medium to large, dense to sparse, wellbehaved to ill-conditioned ones, where the standard method typically used is Davidson's diagonalization. Numerical tests show that, while Davidson's method remains the best choice for most applications in quantum chemistry, LOBPCG represents a competitive alternative, especially when memory is an issue, and can even outperform Davidson for ill-conditioned, non diagonally dominant problems.
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We propose and analyze an approximate message passing (AMP) algorithm for the matrix tensor product model, which is a generalization of the standard spiked matrix models that allows for multiple types of pairwise observations over a collection of latent variables. A key innovation for this algorithm is a method for optimally weighing and combining multiple estimates in each iteration. Building upon an AMP convergence theorem for non-separable functions, we prove a state evolution for non-separable functions that provides an asymptotically exact description of its performance in the high-dimensional limit. We leverage this state evolution result to provide necessary and sufficient conditions for recovery of the signal of interest. Such conditions depend on the singular values of a linear operator derived from an appropriate generalization of a signal-to-noise ratio for our model. Our results recover as special cases a number of recently proposed methods for contextual models (e.g., covariate assisted clustering) as well as inhomogeneous noise models.
We consider the problem of sequentially optimizing a time-varying objective function using time-varying Bayesian optimization (TVBO). Here, the key challenge is the exploration-exploitation trade-off under time variations. Current approaches to TVBO require prior knowledge of a constant rate of change. However, in practice, the rate of change is usually unknown. We propose an event-triggered algorithm, ET-GP-UCB, that treats the optimization problem as static until it detects changes in the objective function online and then resets the dataset. This allows the algorithm to adapt to realized temporal changes without the need for prior knowledge. The event-trigger is based on probabilistic uniform error bounds used in Gaussian process regression. We provide regret bounds for ET-GP-UCB and show in numerical experiments that it outperforms state-of-the-art algorithms on synthetic and real-world data. Furthermore, these results demonstrate that ET-GP-UCB is readily applicable to various settings without tuning hyperparameters.
We study the problem of locally private mean estimation of high-dimensional vectors in the Euclidean ball. Existing algorithms for this problem either incur sub-optimal error or have high communication and/or run-time complexity. We propose a new algorithmic framework, ProjUnit, for private mean estimation that yields algorithms that are computationally efficient, have low communication complexity, and incur optimal error up to a $1+o(1)$-factor. Our framework is deceptively simple: each randomizer projects its input to a random low-dimensional subspace, normalizes the result, and then runs an optimal algorithm such as PrivUnitG in the lower-dimensional space. In addition, we show that, by appropriately correlating the random projection matrices across devices, we can achieve fast server run-time. We mathematically analyze the error of the algorithm in terms of properties of the random projections, and study two instantiations. Lastly, our experiments for private mean estimation and private federated learning demonstrate that our algorithms empirically obtain nearly the same utility as optimal ones while having significantly lower communication and computational cost.
The general adversary dual is a powerful tool in quantum computing because it gives a query-optimal bounded-error quantum algorithm for deciding any Boolean function. Unfortunately, the algorithm uses linear qubits in the worst case, and only works if the constraints of the general adversary dual are exactly satisfied. The challenge of improving the algorithm is that it is brittle to arbitrarily small errors since it relies on a reflection over a span of vectors. We overcome this challenge and build a robust dual adversary algorithm that can handle approximately satisfied constraints. As one application of our robust algorithm, we prove that for any Boolean function with polynomially many 1-valued inputs (or in fact a slightly weaker condition) there is a query-optimal algorithm that uses logarithmic qubits. As another application, we prove that numerically derived, approximate solutions to the general adversary dual give a bounded-error quantum algorithm under certain conditions. Further, we show that these conditions empirically hold with reasonable iterations for Boolean functions with small domains. We also develop several tools that may be of independent interest, including a robust approximate spectral gap lemma, a method to compress a general adversary dual solution using the Johnson-Lindenstrauss lemma, and open-source code to find solutions to the general adversary dual.
Separating signals from an additive mixture may be an unnecessarily hard problem when one is only interested in specific properties of a given signal. In this work, we tackle simpler "statistical component separation" problems that focus on recovering a predefined set of statistical descriptors of a target signal from a noisy mixture. Assuming access to samples of the noise process, we investigate a method devised to match the statistics of the solution candidate corrupted by noise samples with those of the observed mixture. We first analyze the behavior of this method using simple examples with analytically tractable calculations. Then, we apply it in an image denoising context employing 1) wavelet-based descriptors, 2) ConvNet-based descriptors on astrophysics and ImageNet data. In the case of 1), we show that our method better recovers the descriptors of the target data than a standard denoising method in most situations. Additionally, despite not constructed for this purpose, it performs surprisingly well in terms of peak signal-to-noise ratio on full signal reconstruction. In comparison, representation 2) appears less suitable for image denoising. Finally, we extend this method by introducing a diffusive stepwise algorithm which gives a new perspective to the initial method and leads to promising results for image denoising under specific circumstances.
Consider a random sample $(X_{1},\ldots,X_{n})$ from an unknown discrete distribution $P=\sum_{j\geq1}p_{j}\delta_{s_{j}}$ on a countable alphabet $\mathbb{S}$, and let $(Y_{n,j})_{j\geq1}$ be the empirical frequencies of distinct symbols $s_{j}$'s in the sample. We consider the problem of estimating the $r$-order missing mass, which is a discrete functional of $P$ defined as $$\theta_{r}(P;\mathbf{X}_{n})=\sum_{j\geq1}p^{r}_{j}I(Y_{n,j}=0).$$ This is generalization of the missing mass whose estimation is a classical problem in statistics, being the subject of numerous studies both in theory and methods. First, we introduce a nonparametric estimator of $\theta_{r}(P;\mathbf{X}_{n})$ and a corresponding non-asymptotic confidence interval through concentration properties of $\theta_{r}(P;\mathbf{X}_{n})$. Then, we investigate minimax estimation of $\theta_{r}(P;\mathbf{X}_{n})$, which is the main contribution of our work. We show that minimax estimation is not feasible over the class of all discrete distributions on $\mathbb{S}$, and not even for distributions with regularly varying tails, which only guarantee that our estimator is consistent for $\theta_{r}(P;\mathbf{X}_{n})$. This leads to introduce the stronger assumption of second-order regular variation for the tail behaviour of $P$, which is proved to be sufficient for minimax estimation of $\theta_r(P;\mathbf{X}_{n})$, making the proposed estimator an optimal minimax estimator of $\theta_{r}(P;\mathbf{X}_{n})$. Our interest in the $r$-order missing mass arises from forensic statistics, where the estimation of the $2$-order missing mass appears in connection to the estimation of the likelihood ratio $T(P,\mathbf{X}_{n})=\theta_{1}(P;\mathbf{X}_{n})/\theta_{2}(P;\mathbf{X}_{n})$, known as the "fundamental problem of forensic mathematics". We present theoretical guarantees to nonparametric estimation of $T(P,\mathbf{X}_{n})$.
We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions for the $L^p$-error with arbitrary $p \in [1,\infty)$ and for linear functionals of the empirical OT map, together with their moment convergence. The former has a non-Gaussian limit, whose explicit density is derived, while the latter attains asymptotic normality. For both cases, we also establish consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. We also discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation.
Entropic optimal transport (EOT) presents an effective and computationally viable alternative to unregularized optimal transport (OT), offering diverse applications for large-scale data analysis. In this work, we derive novel statistical bounds for empirical plug-in estimators of the EOT cost and show that their statistical performance in the entropy regularization parameter $\epsilon$ and the sample size $n$ only depends on the simpler of the two probability measures. For instance, under sufficiently smooth costs this yields the parametric rate $n^{-1/2}$ with factor $\epsilon^{-d/2}$, where $d$ is the minimum dimension of the two population measures. This confirms that empirical EOT also adheres to the lower complexity adaptation principle, a hallmark feature only recently identified for unregularized OT. As a consequence of our theory, we show that the empirical entropic Gromov-Wasserstein distance and its unregularized version for measures on Euclidean spaces also obey this principle. Additionally, we comment on computational aspects and complement our findings with Monte Carlo simulations. Our techniques employ empirical process theory and rely on a dual formulation of EOT over a single function class. Crucial to our analysis is the observation that the entropic cost-transformation of a function class does not increase its uniform metric entropy by much.
We propose a model to flexibly estimate joint tail properties by exploiting the convergence of an appropriately scaled point cloud onto a compact limit set. Characteristics of the shape of the limit set correspond to key tail dependence properties. We directly model the shape of the limit set using B\'ezier splines, which allow flexible and parsimonious specification of shapes in two dimensions. We then fit the B\'ezier splines to data in pseudo-polar coordinates using Markov chain Monte Carlo, utilizing a limiting approximation to the conditional likelihood of the radii given angles. By imposing appropriate constraints on the parameters of the B\'ezier splines, we guarantee that each posterior sample is a valid limit set boundary, allowing direct posterior analysis of any quantity derived from the shape of the curve. Furthermore, we obtain interpretable inference on the asymptotic dependence class by using mixture priors with point masses on the corner of the unit box. Finally, we apply our model to bivariate datasets of extremes of variables related to fire risk and air pollution.
The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad--Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the model.
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