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We consider a high-dimensional mean estimation problem over a binary hidden Markov model, which illuminates the interplay between memory in data, sample size, dimension, and signal strength in statistical inference. In this model, an estimator observes $n$ samples of a $d$-dimensional parameter vector $\theta_{*}\in\mathbb{R}^{d}$, multiplied by a random sign $ S_i $ ($1\le i\le n$), and corrupted by isotropic standard Gaussian noise. The sequence of signs $\{S_{i}\}_{i\in[n]}\in\{-1,1\}^{n}$ is drawn from a stationary homogeneous Markov chain with flip probability $\delta\in[0,1/2]$. As $\delta$ varies, this model smoothly interpolates two well-studied models: the Gaussian Location Model for which $\delta=0$ and the Gaussian Mixture Model for which $\delta=1/2$. Assuming that the estimator knows $\delta$, we establish a nearly minimax optimal (up to logarithmic factors) estimation error rate, as a function of $\|\theta_{*}\|,\delta,d,n$. We then provide an upper bound to the case of estimating $\delta$, assuming a (possibly inaccurate) knowledge of $\theta_{*}$. The bound is proved to be tight when $\theta_{*}$ is an accurately known constant. These results are then combined to an algorithm which estimates $\theta_{*}$ with $\delta$ unknown a priori, and theoretical guarantees on its error are stated.

Drawing a direct analogy with the well-studied vibration or elastic modes, we introduce an object's fracture modes, which constitute its preferred or most natural ways of breaking. We formulate a sparsified eigenvalue problem, which we solve iteratively to obtain the n lowest-energy modes. These can be precomputed for a given shape to obtain a prefracture pattern that can substitute the state of the art for realtime applications at no runtime cost but significantly greater realism. Furthermore, any realtime impact can be projected onto our modes to obtain impact-dependent fracture patterns without the need for any online crack propagation simulation. We not only introduce this theoretically novel concept, but also show its fundamental and practical superiority in a diverse set of examples and contexts.

This paper investigates the problem of regret minimization in linear time-varying (LTV) dynamical systems. Due to the simultaneous presence of uncertainty and non-stationarity, designing online control algorithms for unknown LTV systems remains a challenging task. At a cost of NP-hard offline planning, prior works have introduced online convex optimization algorithms, although they suffer from nonparametric rate of regret. In this paper, we propose the first computationally tractable online algorithm with regret guarantees that avoids offline planning over the state linear feedback policies. Our algorithm is based on the optimism in the face of uncertainty (OFU) principle in which we optimistically select the best model in a high confidence region. Our algorithm is then more explorative when compared to previous approaches. To overcome non-stationarity, we propose either a restarting strategy (R-OFU) or a sliding window (SW-OFU) strategy. With proper configuration, our algorithm is attains sublinear regret $O(T^{2/3})$. These algorithms utilize data from the current phase for tracking variations on the system dynamics. We corroborate our theoretical findings with numerical experiments, which highlight the effectiveness of our methods. To the best of our knowledge, our study establishes the first model-based online algorithm with regret guarantees under LTV dynamical systems.

Analyzing time series in the frequency domain enables the development of powerful tools for investigating the second-order characteristics of multivariate stochastic processes. Parameters like the spectral density matrix and its inverse, the coherence or the partial coherence, encode comprehensively the complex linear relations between the component processes of the multivariate system. In this paper, we develop inference procedures for such parameters in a high-dimensional, time series setup. In particular, we first focus on the derivation of consistent estimators of the coherence and, more importantly, of the partial coherence which possess manageable limiting distributions that are suitable for testing purposes. Statistical tests of the hypothesis that the maximum over frequencies of the coherence, respectively, of the partial coherence, do not exceed a prespecified threshold value are developed. Our approach allows for testing hypotheses for individual coherences and/or partial coherences as well as for multiple testing of large sets of such parameters. In the latter case, a consistent procedure to control the false discovery rate is developed. The finite sample performance of the inference procedures proposed is investigated by means of simulations and applications to the construction of graphical interaction models for brain connectivity based on EEG data are presented.

In ordinary Dimensionality Reduction (DR), each data instance in a high dimensional space (original space), or on a distance matrix denoting original space distances, is mapped to (projected onto) one point in a low dimensional space (visual space), building a layout of projected points trying to preserve as much as possible some property of data such as distances, neighbourhood relationships, and/or topology structures, with the ultimate goal of approximating semantic properties of data with preserved geometric properties or topology structures in visual space. In this paper, the concept of Multi-point Dimensionality Reduction is elaborated on where each data instance can be mapped to (projected onto) possibly more than one point in visual space by providing the first general solution (algorithm) for it as a move in the direction of improving reliablity, usability and interpretability of dimensionality reduction. Furthermore by allowing the points in visual space to be split into two layers while maintaining the possibility of having more than one projection (mapping) per data instance , the benefit of separating more reliable points from less reliable points is dicussed notwithstanding the effort to improve less reliable points. The proposed solution (algorithm) in this paper, named Layered Vertex Splitting Data Embedding (LVSDE), is built upon and extends a combination of ordinary DR and graph drawing techniques. Based on the experiments of this paper on some data sets, the particular proposed algorithm (LVSDE) practically outperforms popular ordinary DR methods visually (semantics, group separation, subgroup detection or combinational group detection) in a way that is easily explainable.

We report an improvement to the conventional Echo State Network (ESN), which already achieves competitive performance in one-dimensional time series prediction of dynamical systems. Our model -- a 20$\%$-dense ESN with reservoir weights derived from a fruit fly connectome (and from its bootstrapped distribution) -- yields superior performance on a chaotic time series prediction task, and furthermore alleviates the ESN's high-variance problem. We also find that an arbitrary positioning of weights can degrade ESN performance and variance; and that this can be remedied in particular by employing connectome-derived weight positions. Herein we consider four connectome features -- namely, the sparsity, positioning, distribution, and clustering of weights -- and construct corresponding model classes (A, B, B${}_2$, C) from an appropriate null model ESN; one with its reservoir layer replaced by a fruit fly connectivity matrix. After tuning relevant hyperparameters and selecting the best instance of each model class, we train and validate all models for multi-step prediction on size-variants (50, 250, 500, and 750 training input steps) of the Mackey-Glass chaotic time series; and compute their performance (Mean-Squared Error) and variance across train-validate trials.

In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-convex setting is also described. For the sample size $n$, the step size $\eta$, and the batch size $B$, we derive uniform evaluations on the time with orders $n^{-1}$, $\eta^{1/2}$, and $\sqrt{(n - B) / B (n - 1)}$, respectively.

Machine learning approaches commonly rely on the assumption of independent and identically distributed (i.i.d.) data. In reality, however, this assumption is almost always violated due to distribution shifts between environments. Although valuable learning signals can be provided by heterogeneous data from changing distributions, it is also known that learning under arbitrary (adversarial) changes is impossible. Causality provides a useful framework for modeling distribution shifts, since causal models encode both observational and interventional distributions. In this work, we explore the sparse mechanism shift hypothesis, which posits that distribution shifts occur due to a small number of changing causal conditionals. Motivated by this idea, we apply it to learning causal structure from heterogeneous environments, where i.i.d. data only allows for learning an equivalence class of graphs without restrictive assumptions. We propose the Mechanism Shift Score (MSS), a score-based approach amenable to various empirical estimators, which provably identifies the entire causal structure with high probability if the sparse mechanism shift hypothesis holds. Empirically, we verify behavior predicted by the theory and compare multiple estimators and score functions to identify the best approaches in practice. Compared to other methods, we show how MSS bridges a gap by both being nonparametric as well as explicitly leveraging sparse changes.

The synthetic control method has become a widely popular tool to estimate causal effects with observational data. Despite this, inference for synthetic control methods remains challenging. Often, inferential results rely on linear factor model data generating processes. In this paper, we characterize the conditions on the factor model primitives (the factor loadings) for which the statistical risk minimizers are synthetic controls (in the simplex). Then, we propose a Bayesian alternative to the synthetic control method that preserves the main features of the standard method and provides a new way of doing valid inference. We explore a Bernstein-von Mises style result to link our Bayesian inference to the frequentist inference. For linear factor model frameworks we show that a maximum likelihood estimator (MLE) of the synthetic control weights can consistently estimate the predictive function of the potential outcomes for the treated unit and that our Bayes estimator is asymptotically close to the MLE in the total variation sense. Through simulations, we show that there is convergence between the Bayes and frequentist approach even in sparse settings. Finally, we apply the method to re-visit the study of the economic costs of the German re-unification. The Bayesian synthetic control method is available in the bsynth R-package.

In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques.