We propose the first near-optimal quantum algorithm for estimating in Euclidean norm the mean of a vector-valued random variable with finite mean and covariance. Our result aims at extending the theory of multivariate sub-Gaussian estimators to the quantum setting. Unlike classically, where any univariate estimator can be turned into a multivariate estimator with at most a logarithmic overhead in the dimension, no similar result can be proved in the quantum setting. Indeed, Heinrich ruled out the existence of a quantum advantage for the mean estimation problem when the sample complexity is smaller than the dimension. Our main result is to show that, outside this low-precision regime, there is a quantum estimator that outperforms any classical estimator. Our approach is substantially more involved than in the univariate setting, where most quantum estimators rely only on phase estimation. We exploit a variety of additional algorithmic techniques such as amplitude amplification, the Bernstein-Vazirani algorithm, and quantum singular value transformation. Our analysis also uses concentration inequalities for multivariate truncated statistics. We develop our quantum estimators in two different input models that showed up in the literature before. The first one provides coherent access to the binary representation of the random variable and it encompasses the classical setting. In the second model, the random variable is directly encoded into the phases of quantum registers. This model arises naturally in many quantum algorithms but it is often incomparable to having classical samples. We adapt our techniques to these two settings and we show that the second model is strictly weaker for solving the mean estimation problem. Finally, we describe several applications of our algorithms, notably in measuring the expectation values of commuting observables and in the field of machine learning.
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Much of the literature on optimal design of bandit algorithms is based on minimization of expected regret. It is well known that designs that are optimal over certain exponential families can achieve expected regret that grows logarithmically in the number of arm plays, at a rate governed by the Lai-Robbins lower bound. In this paper, we show that when one uses such optimized designs, the regret distribution of the associated algorithms necessarily has a very heavy tail, specifically, that of a truncated Cauchy distribution. Furthermore, for $p>1$, the $p$'th moment of the regret distribution grows much faster than poly-logarithmically, in particular as a power of the total number of arm plays. We show that optimized UCB bandit designs are also fragile in an additional sense, namely when the problem is even slightly mis-specified, the regret can grow much faster than the conventional theory suggests. Our arguments are based on standard change-of-measure ideas, and indicate that the most likely way that regret becomes larger than expected is when the optimal arm returns below-average rewards in the first few arm plays, thereby causing the algorithm to believe that the arm is sub-optimal. To alleviate the fragility issues exposed, we show that UCB algorithms can be modified so as to ensure a desired degree of robustness to mis-specification. In doing so, we also provide a sharp trade-off between the amount of UCB exploration and the tail exponent of the resulting regret distribution.
Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used $K$-means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the Euclidean space, centroid-based and distance-based formulations of the $K$-means are equivalent. In modern machine learning applications, data often arise as probability distributions and a natural generalization to handle measure-valued data is to use the optimal transport metric. Due to non-negative Alexandrov curvature of the Wasserstein space, barycenters suffer from regularity and non-robustness issues. The peculiar behaviors of Wasserstein barycenters may make the centroid-based formulation fail to represent the within-cluster data points, while the more direct distance-based $K$-means approach and its semidefinite program (SDP) relaxation are capable of recovering the true cluster labels. In the special case of clustering Gaussian distributions, we show that the SDP relaxed Wasserstein $K$-means can achieve exact recovery given the clusters are well-separated under the $2$-Wasserstein metric. Our simulation and real data examples also demonstrate that distance-based $K$-means can achieve better classification performance over the standard centroid-based $K$-means for clustering probability distributions and images.
We consider a potential outcomes model in which interference may be present between any two units but the extent of interference diminishes with spatial distance. The causal estimand is the global average treatment effect, which compares outcomes under the counterfactuals that all or no units are treated. We study a class of designs in which space is partitioned into clusters that are randomized into treatment and control. For each design, we estimate the treatment effect using a Horvitz-Thompson estimator that compares the average outcomes of units with all or no neighbors treated, where the neighborhood radius is of the same order as the cluster size dictated by the design. We derive the estimator's rate of convergence as a function of the design and degree of interference and use this to obtain estimator-design pairs that achieve near-optimal rates of convergence under relatively minimal assumptions on interference. We prove that the estimators are asymptotically normal and provide a variance estimator. For practical implementation of the designs, we suggest partitioning space using clustering algorithms.
Computing Wasserstein barycenters of discrete measures has recently attracted considerable attention due to its wide variety of applications in data science. In general, this problem is NP-hard, calling for practical approximative algorithms. In this paper, we analyze a well-known simple framework for approximating Wasserstein-$p$ barycenters, where we mainly consider the most common case $p=2$ and $p=1$, which is not as well discussed. The framework produces sparse support solutions and shows good numerical results in the free-support setting. Depending on the desired level of accuracy, this requires only $N-1$ or $N(N-1)/2$ standard two-marginal optimal transport (OT) computations between the $N$ input measures, respectively, which is fast, memory-efficient and easy to implement using any OT solver as a black box. What is more, these methods yield a relative error of at most $N$ and $2$, respectively, for both $p=1, 2$. We show that these bounds are practically sharp. In light of the hardness of the problem, it is not surprising that such guarantees cannot be close to optimality in general. Nevertheless, these error bounds usually turn out to be drastically lower for a given particular problem in practice and can be evaluated with almost no computational overhead, in particular without knowledge of the optimal solution. In our numerical experiments, this guaranteed errors of at most a few percent.
In this paper, a new regularization term is proposed to solve mathematical image problems. By using difference operators in the four directions; horizontal, vertical and two diagonal directions, an estimation of derivative amplitude is found. Based on the new obtained estimation, a new regularization term will be defined, which can be viewed as a new discretized total variation (TVprn) model. By improving TVprn, a more effective regularization term is introduced. By finding conjugate of TVprn and producing vector fields with special constraints, a new discretized TV for two dimensional discrete functions is proposed (TVnew). The capability of the new TV model to solve mathematical image problems is examined in some numerical experiments. It is shown that the new proposed TV model can reconstruct the edges and corners of the noisy images better than other TVs. Moreover, two test experiments of resolution enhancement problem are solved and compared with some other different TVs.
We prove that the partition rank and the analytic rank of tensors are equal up to a constant, over finite fields of any characteristic and any large enough cardinality depending on the analytic rank. Moreover, we show that a plausible improvement of our field cardinality requirement would imply that the ranks are equal up to 1+o(1) in the exponent over every finite field. At the core of the proof is a technique for lifting decompositions of multilinear polynomials in an open subset of an algebraic variety, and a technique for finding a large subvariety that retains all rational points such that at least one of these points satisfies a finite-field analogue of genericity with respect to it. Proving the equivalence between these two ranks, ideally over fixed finite fields, is a central question in additive combinatorics, and was reiterated by multiple authors. As a corollary we prove, allowing the field to depend on the value of the norm, the Polynomial Gowers Inverse Conjecture in the d vs. d-1 case.
The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently we provided an upper bound on the quantum mixing time for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain.
There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample $x$, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in $ [0,1]$ acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near $1$, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near $0$, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.
We study the problem of estimating an unknown parameter in a distributed and online manner. Existing work on distributed online learning typically either focuses on asymptotic analysis, or provides bounds on regret. However, these results may not directly translate into bounds on the error of the learned model after a finite number of time-steps. In this paper, we propose a distributed online estimation algorithm which enables each agent in a network to improve its estimation accuracy by communicating with neighbors. We provide non-asymptotic bounds on the estimation error, leveraging the statistical properties of the underlying model. Our analysis demonstrates a trade-off between estimation error and communication costs. Further, our analysis allows us to determine a time at which the communication can be stopped (due to the costs associated with communications), while meeting a desired estimation accuracy. We also provide a numerical example to validate our results.
The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its simplest "symmetric" form, it asserts that whenever a bad-event has probability $p$ and affects at most $d$ bad-events, and $e p d < 1$, then a configuration avoiding all $\mathcal B$ exists. A seminal algorithm of Moser & Tardos (2010) gives nearly-automatic randomized algorithms for most constructions based on the LLL. However, deterministic algorithms have lagged behind. We address three specific shortcomings of the prior deterministic algorithms. First, our algorithm applies to the LLL criterion of Shearer (1985); this is more powerful than alternate LLL criteria and also removes a number of nuisance parameters and leads to cleaner and more legible bounds. Second, we provide parallel algorithms with much greater flexibility in the functional form of of the bad-events. Third, we provide a derandomized version of the MT-distribution, that is, the distribution of the variables at the termination of the MT algorithm. We show applications to non-repetitive vertex coloring, independent transversals, strong coloring, and other problems. These give deterministic algorithms which essentially match the best previous randomized sequential and parallel algorithms.
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