We develop novel methods for using persistent homology to infer the homology of an unknown Riemannian manifold $(M, g)$ from a point cloud sampled from an arbitrary smooth probability density function. Standard distance-based filtered complexes, such as the \v{C}ech complex, often have trouble distinguishing noise from features that are simply small. We address this problem by defining a family of "density-scaled filtered complexes" that includes a density-scaled \v{C}ech complex and a density-scaled Vietoris--Rips complex. We show that the density-scaled \v{C}ech complex is homotopy-equivalent to $M$ for filtration values in an interval whose starting point converges to $0$ in probability as the number of points $N \to \infty$ and whose ending point approaches infinity as $N \to \infty$. By contrast, the standard \v{C}ech complex may only be homotopy-equivalent to $M$ for a very small range of filtration values. The density-scaled filtered complexes also have the property that they are invariant under conformal transformations, such as scaling. We implement a filtered complex $\widehat{DVR}$ that approximates the density-scaled Vietoris--Rips complex, and we empirically test the performance of our implementation. As examples, we use $\widehat{DVR}$ to identify clusters that have different densities, and we apply $\widehat{DVR}$ to a time-delay embedding of the Lorenz dynamical system. Our implementation is stable (under conditions that are almost surely satisfied) and designed to handle outliers in the point cloud that do not lie on $M$.
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Given its status as a classic problem and its importance to both theoreticians and practitioners, edit distance provides an excellent lens through which to understand how the theoretical analysis of algorithms impacts practical implementations. From an applied perspective, the goals of theoretical analysis are to predict the empirical performance of an algorithm and to serve as a yardstick to design novel algorithms that perform well in practice. In this paper, we systematically survey the types of theoretical analysis techniques that have been applied to edit distance and evaluate the extent to which each one has achieved these two goals. These techniques include traditional worst-case analysis, worst-case analysis parametrized by edit distance or entropy or compressibility, average-case analysis, semi-random models, and advice-based models. We find that the track record is mixed. On one hand, two algorithms widely used in practice have been born out of theoretical analysis and their empirical performance is captured well by theoretical predictions. On the other hand, all the algorithms developed using theoretical analysis as a yardstick since then have not had any practical relevance. We conclude by discussing the remaining open problems and how they can be tackled.
Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-\epsilon$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $\alpha$-locally differentially private mechanism. The remaining $\epsilon n $ batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be $\epsilon/\sqrt{k}+\sqrt{d/kn}$, up to a $\sqrt{\log(1/\epsilon)}$ factor. Under the privacy constraint alone, the minimax rate of estimation is $\sqrt{d^2/\alpha^2 kn}$. We show that combining the two constraints leads to a minimax estimation rate of $\epsilon\sqrt{d/\alpha^2 k}+\sqrt{d^2/\alpha^2 kn}$ up to a $\sqrt{\log(1/\epsilon)}$ factor, larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.
We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.
Heavy ball momentum is a popular acceleration idea in stochastic optimization. There have been several attempts to understand its perceived benefits, but the complete picture is still unclear. Specifically, the error expression in the presence of noise has two separate terms: the bias and the variance, but most existing works only focus on bias and show that momentum accelerates its decay. Such analyses overlook the interplay between bias and variance and, therefore, miss important implications. In this work, we analyze a sample complexity bound of stochastic approximation algorithms with heavy-ball momentum that accounts for both bias and variance. We find that for the same step size, which is small enough, the iterates with momentum have improved sample complexity compared to the ones without. However, by using a different step-size sequence, the non-momentum version can nullify this benefit. Subsequently, we show that our sample complexity bounds are indeed tight for a small enough neighborhood around the solution and large enough noise variance. Our analysis also sheds some light on the finite-time behavior of these algorithms. This explains the perceived benefit in the initial phase of momentum-based schemes.
To simulate noisy boson sampling approximating it by only the lower-order multi-boson interferences (e.g., by a smaller number of interfering bosons and classical particles) is very popular idea. I show that the output data from any such classical simulations can be efficiently distinguished from that of the quantum device they try to simulate, even with finite noise in the latter. The distinguishing datasets can be the experimental estimates of some large probabilities, a wide class of such is presented. This is a sequel of \textit{Quantum} \textbf{5}, 423 (2021), where I present more accessible account of the main result enhanced by additional insight on the contribution from the higher-order multi-boson interferences in presence of noise.
Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.
We introduce a novel methodology for particle filtering in dynamical systems where the evolution of the signal of interest is described by a SDE and observations are collected instantaneously at prescribed time instants. The new approach includes the discretisation of the SDE and the design of efficient particle filters for the resulting discrete-time state-space model. The discretisation scheme converges with weak order 1 and it is devised to create a sequential dependence structure along the coordinates of the discrete-time state vector. We introduce a class of space-sequential particle filters that exploits this structure to improve performance when the system dimension is large. This is numerically illustrated by a set of computer simulations for a stochastic Lorenz 96 system with additive noise. The new space-sequential particle filters attain approximately constant estimation errors as the dimension of the Lorenz 96 system is increased, with a computational cost that increases polynomially, rather than exponentially, with the system dimension. Besides the new numerical scheme and particle filters, we provide in this paper a general framework for discrete-time filtering in continuous-time dynamical systems described by a SDE and instantaneous observations. Provided that the SDE is discretised using a weakly-convergent scheme, we prove that the marginal posterior laws of the resulting discrete-time state-space model converge to the posterior marginal posterior laws of the original continuous-time state-space model under a suitably defined metric. This result is general and not restricted to the numerical scheme or particle filters specifically studied in this manuscript.
The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.
Universal coding of integers~(UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1,H(P)\}$ is within a constant factor, where $H(P)$ is the Shannon entropy of the decreasing probability distribution $P$. However, if we consider the ratio of the expected codeword length to $H(P)$, the ratio tends to infinity by using UCI, when $H(P)$ tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized universal coding of integers~(GUCI), such that the ratio of the expected codeword length to $H(P)$ is within a constant factor $K$. First, the definition of GUCI is proposed and the coding structure of GUCI is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the range $1\leq K_{\mathcal{C}}^{*}\leq 2$. Then, by comparing UCI and GUCI, we show that when the entropy is very large or $P(0)$ is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.
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