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Recently, high dimensional vector auto-regressive models (VAR), have attracted a lot of interest, due to novel applications in the health, engineering and social sciences. The presence of temporal dependence poses additional challenges to the theory of penalized estimation techniques widely used in the analysis of their iid counterparts. However, recent work (e.g., [Basu and Michailidis, 2015, Kock and Callot, 2015]) has established optimal consistency of $\ell_1$-LASSO regularized estimates applied to models involving high dimensional stable, Gaussian processes. The only price paid for temporal dependence is an extra multiplicative factor that equals 1 for independent and identically distributed (iid) data. Further, [Wong et al., 2020] extended these results to heavy tailed VARs that exhibit "$\beta$-mixing" dependence, but the rates rates are sub-optimal, while the extra factor is intractable. This paper improves these results in two important directions: (i) We establish optimal consistency rates and corresponding finite sample bounds for the underlying model parameters that match those for iid data, modulo a price for temporal dependence, that is easy to interpret and equals 1 for iid data. (ii) We incorporate more general penalties in estimation (which are not decomposable unlike the $\ell_1$ norm) to induce general sparsity patterns. The key technical tool employed is a novel, easy-to-use concentration bound for heavy tailed linear processes, that do not rely on "mixing" notions and give tighter bounds.

The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Number, MAX-$k$-SAT, and Set Cover. In this paper, we show that fine-grained reductions implying even $\lambda^n$-hardness of these problems from SETH for any $\lambda>1$, would imply new circuit lower bounds: super-linear lower bounds for Boolean series-parallel circuits or polynomial lower bounds for arithmetic circuits (each of which is a four-decade open question). We also extend this barrier result to the class of parameterized problems. Namely, for every $\lambda>1$ we conditionally rule out fine-grained reductions implying SETH-based lower bounds of $\lambda^k$ for a number of problems parameterized by the solution size $k$. Our main technical tool is a new concept called polynomial formulations. In particular, we show that many problems can be represented by relatively succinct low-degree polynomials, and that any problem with such a representation cannot be proven SETH-hard (without proving new circuit lower bounds).

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schr\"odinger framework to solve the Liouville equation for the HJE, since it can be recast as the semiclassical limit of the Wigner transform of the Schr\"odinger equation. Comparsion between the Schr\"odinger and the Liouville framework will also be made.

The paper studies non-stationary high-dimensional vector autoregressions of order $k$, VAR($k$). Additional deterministic terms such as trend or seasonality are allowed. The number of time periods, $T$, and number of coordinates, $N$, are assumed to be large and of the same order. Under such regime the first-order asymptotics of the Johansen likelihood ratio (LR), Pillai-Barlett, and Hotelling-Lawley tests for cointegration is derived: Test statistics converge to non-random integrals. For more refined analysis, the paper proposes and analyzes a modification of the Johansen test. The new test for the absence of cointegration converges to the partial sum of the Airy$_1$ point process. Supporting Monte Carlo simulations indicate that the same behavior persists universally in many situations beyond our theorems. The paper presents an empirical implementation of the approach to the analysis of stocks in S$\&$P$100$ and of cryptocurrencies. The latter example has strong presence of multiple cointegrating relationships, while the former is consistent with the null of no cointegration.

In data science, vector autoregression (VAR) models are popular in modeling multivariate time series in the environmental sciences and other applications. However, these models are computationally complex with the number of parameters scaling quadratically with the number of time series. In this work, we propose a so-called neighborhood vector autoregression (NVAR) model to efficiently analyze large-dimensional multivariate time series. We assume that the time series have underlying neighborhood relationships, e.g., spatial or network, among them based on the inherent setting of the problem. When this neighborhood information is available or can be summarized using a distance matrix, we demonstrate that our proposed NVAR method provides a computationally efficient and theoretically sound estimation of model parameters. The performance of the proposed method is compared with other existing approaches in both simulation studies and a real application of stream nitrogen study.

Modeling and control of high-dimensional, nonlinear robotic systems remains a challenging task. While various model- and learning-based approaches have been proposed to address these challenges, they broadly lack generalizability to different control tasks and rarely preserve the structure of the dynamics. In this work, we propose a new, data-driven approach for extracting low-dimensional models from data using Spectral Submanifold Reduction (SSMR). In contrast to other data-driven methods which fit dynamical models to training trajectories, we identify the dynamics on generic, low-dimensional attractors embedded in the full phase space of the robotic system. This allows us to obtain computationally-tractable models for control which preserve the system's dominant dynamics and better track trajectories radically different from the training data. We demonstrate the superior performance and generalizability of SSMR in dynamic trajectory tracking tasks vis-a-vis the state of the art, including Koopman operator-based approaches.

As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and relative difficulty of interpretation. This paper introduces a novel infinite-order VAR model which, with only a little sacrifice of generality, inherits the essential temporal patterns of the VARMA model but avoids all of the above drawbacks. As another attractive feature, the temporal and cross-sectional dependence structures of this model can be interpreted separately, since they are characterized by different sets of parameters. For high-dimensional time series, this separation motivates us to impose sparsity on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved, while no loss of temporal information is incurred by the imposed sparsity. We introduce an $\ell_1$-regularized estimator for the proposed model and derive the corresponding nonasymptotic error bounds. An efficient block coordinate descent algorithm and a consistent model order selection method are developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis.

Bayesian variable selection methods are powerful techniques for fitting and inferring on sparse high-dimensional linear regression models. However, many are computationally intensive or require restrictive prior distributions on model parameters. Likelihood based penalization methods are more computationally friendly, but resource intensive refitting techniques are needed for inference. In this paper, we proposed an efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are required through the use of plug-in empirical Bayes estimates of hyperparameters. Efficient maximum a posteriori probability (MAP) estimation is completed through the use of a partitioned and extended expectation conditional maximization (ECM) algorithm. The result is a PaRtitiOned empirical Bayes Ecm (PROBE) algorithm applied to sparse high-dimensional linear regression. We propose methods to estimate credible and prediction intervals for predictions of future values. We compare the empirical properties of predictions and our predictive inference to comparable approaches with numerous simulation studies and an analysis of cancer cell lines drug response study. The proposed approach is implemented in the R package probe.

The edges of the characteristic imset polytope, $\operatorname{CIM}_p$, were recently shown to have strong connections to causal discovery as many algorithms could be interpreted as greedy restricted edge-walks, even though only a strict subset of the edges are known. To better understand the general edge structure of the polytope we describe the edge structure of faces with a clear combinatorial interpretation: for any undirected graph $G$ we have the face $\operatorname{CIM}_G$, the convex hull of the characteristic imsets of DAGs with skeleton $G$. We give a full edge-description of $\operatorname{CIM}_G$ when $G$ is a tree, leading to interesting connections to other polytopes. In particular the well-studied stable set polytope can be recovered as a face of $\operatorname{CIM}_G$ when $G$ is a tree. Building on this connection we are also able to give a description of all edges of $\operatorname{CIM}_G$ when $G$ is a cycle, suggesting possible inroads for generalization. We then introduce an algorithm for learning directed trees from data, utilizing our newly discovered edges, that outperforms classical methods on simulated Gaussian data.

In recent years, many accelerators have been proposed to efficiently process sparse tensor algebra applications (e.g., sparse neural networks). However, these proposals are single points in a large and diverse design space. The lack of systematic description and modeling support for these sparse tensor accelerators impedes hardware designers from efficient and effective design space exploration. This paper first presents a unified taxonomy to systematically describe the diverse sparse tensor accelerator design space. Based on the proposed taxonomy, it then introduces Sparseloop, the first fast, accurate, and flexible analytical modeling framework to enable early-stage evaluation and exploration of sparse tensor accelerators. Sparseloop comprehends a large set of architecture specifications, including various dataflows and sparse acceleration features (e.g., elimination of zero-based compute). Using these specifications, Sparseloop evaluates a design's processing speed and energy efficiency while accounting for data movement and compute incurred by the employed dataflow as well as the savings and overhead introduced by the sparse acceleration features using stochastic tensor density models. Across representative accelerators and workloads, Sparseloop achieves over 2000 times faster modeling speed than cycle-level simulations, maintains relative performance trends, and achieves 0.1% to 8% average error. With a case study, we demonstrate Sparseloop's ability to help reveal important insights for designing sparse tensor accelerators (e.g., it is important to co-design orthogonal design aspects).