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Running quantum algorithms protected by quantum error correction requires a real time, classical decoder. To prevent the accumulation of a backlog, this decoder must process syndromes from the quantum device at a faster rate than they are generated. Most prior work on real time decoding has focused on an isolated logical qubit encoded in the surface code. However, for surface code, quantum programs of utility will require multi-qubit interactions performed via lattice surgery. A large merged patch can arise during lattice surgery -- possibly as large as the entire device. This puts a significant strain on a real time decoder, which must decode errors on this merged patch and maintain the level of fault-tolerance that it achieves on isolated logical qubits. These requirements are relaxed by using spatially parallel decoding, which can be accomplished by dividing the physical qubits on the device into multiple overlapping groups and assigning a decoder module to each. We refer to this approach as spatially parallel windows. While previous work has explored similar ideas, none have addressed system-specific considerations pertinent to the task or the constraints from using hardware accelerators. In this work, we demonstrate how to configure spatially parallel windows, so that the scheme (1) is compatible with hardware accelerators, (2) supports general lattice surgery operations, (3) maintains the fidelity of the logical qubits, and (4) meets the throughput requirement for real time decoding. Furthermore, our results reveal the importance of optimally choosing the buffer width to achieve a balance between accuracy and throughput -- a decision that should be influenced by the device's physical noise.

We study the properties of a family of distances between functions of a single variable. These distances are examples of integral probability metrics, and have been used previously for comparing probability measures on the line; special cases include the Earth Mover's Distance and the Kolmogorov Metric. We examine their properties for general signals, proving that they are robust to a broad class of deformations. We also establish corresponding robustness results for the induced sliced distances between multivariate functions. Finally, we establish error bounds for approximating the univariate metrics from finite samples, and prove that these approximations are robust to additive Gaussian noise. The results are illustrated in numerical experiments, which include comparisons with Wasserstein distances.

We propose an efficient and easy-to-implement gradient-enhanced least squares Monte Carlo method for computing price and Greeks (i.e., derivatives of the price function) of high-dimensional American options. It employs the sparse Hermite polynomial expansion as a surrogate model for the continuation value function, and essentially exploits the fast evaluation of gradients. The expansion coefficients are computed by solving a linear least squares problem that is enhanced by gradient information of simulated paths. We analyze the convergence of the proposed method, and establish an error estimate in terms of the best approximation error in the weighted $H^1$ space, the statistical error of solving discrete least squares problems, and the time step size. We present comprehensive numerical experiments to illustrate the performance of the proposed method. The results show that it outperforms the state-of-the-art least squares Monte Carlo method with more accurate price, Greeks, and optimal exercise strategies in high dimensions but with nearly identical computational cost, and it can deliver comparable results with recent neural network-based methods up to dimension 100.

We present a new algorithm for imitation learning in infinite horizon linear MDPs dubbed ILARL which greatly improves the bound on the number of trajectories that the learner needs to sample from the environment. In particular, we remove exploration assumptions required in previous works and we improve the dependence on the desired accuracy $\epsilon$ from $\mathcal{O}\br{\epsilon^{-5}}$ to $\mathcal{O}\br{\epsilon^{-4}}$. Our result relies on a connection between imitation learning and online learning in MDPs with adversarial losses. For the latter setting, we present the first result for infinite horizon linear MDP which may be of independent interest. Moreover, we are able to provide a strengthen result for the finite horizon case where we achieve $\mathcal{O}\br{\epsilon^{-2}}$. Numerical experiments with linear function approximation shows that ILARL outperforms other commonly used algorithms.

This thesis is a corpus-based, quantitative, and typological analysis of the functions of Early Slavic participle constructions and their finite competitors ($jegda$-'when'-clauses). The first part leverages detailed linguistic annotation on Early Slavic corpora at the morphosyntactic, dependency, information-structural, and lexical levels to obtain indirect evidence for different potential functions of participle clauses and their main finite competitor and understand the roles of compositionality and default discourse reasoning as explanations for the distribution of participle constructions and $jegda$-clauses in the corpus. The second part uses massively parallel data to analyze typological variation in how languages express the semantic space of English $when$, whose scope encompasses that of Early Slavic participle constructions and $jegda$-clauses. Probabilistic semantic maps are generated and statistical methods (including Kriging, Gaussian Mixture Modelling, precision and recall analysis) are used to induce cross-linguistically salient dimensions from the parallel corpus and to study conceptual variation within the semantic space of the hypothetical concept WHEN.

This paper presents a discriminative classifier for compositional data. This classifier is based on the posterior distribution of the Generalized Dirichlet which is the discriminative counterpart of Generalized Dirichlet mixture model. Moreover, following the mixture of experts paradigm, we proposed a hierarchical mixture of this classifier. In order to learn the models parameters, we use a variational approximation by deriving an upper-bound for the Generalized Dirichlet mixture. To the best of our knownledge, this is the first time this bound is proposed in the literature. Experimental results are presented for spam detection and color space identification.

Optimization over the set of matrices that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods, such as Riemannian approaches, which require a computationally expensive eigenvalue decomposition involving fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to a random estimate of the feasible set. Our method does not enforce the constraint in every iteration exactly, but instead it produces iterations that converge to a critical point on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian counterparts involving the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and GEVP.

We propose a new static program analysis called program behavior analysis. The analysis aims to calculate possible symbolic expressions for every variable at each program point. We design a new lattice, transfer function, and widening operator to accommodate the analysis. Furthermore, we extend the analysis to interprocedural.

Block classical Gram-Schmidt (BCGS) is commonly used for orthogonalizing a set of vectors $X$ in distributed computing environments due to its favorable communication properties relative to other orthogonalization approaches, such as modified Gram-Schmidt or Householder. However, it is known that BCGS (as well as recently developed low-synchronization variants of BCGS) can suffer from a significant loss of orthogonality in finite-precision arithmetic, which can contribute to instability and inaccurate solutions in downstream applications such as $s$-step Krylov subspace methods. A common solution to improve the orthogonality among the vectors is reorthogonalization. Focusing on the "Pythagorean" variant of BCGS, introduced in [E. Carson, K. Lund, & M. Rozlo\v{z}n\'{i}k. SIAM J. Matrix Anal. Appl. 42(3), pp. 1365--1380, 2021], which guarantees an $O(\varepsilon)\kappa^2(X)$ bound on the loss of orthogonality as long as $O(\varepsilon)\kappa^2(X)<1$, where $\varepsilon$ denotes the unit roundoff, we introduce and analyze two reorthogonalized Pythagorean BCGS variants. These variants feature favorable communication properties, with asymptotically two synchronization points per block column, as well as an improved $O(\varepsilon)$ bound on the loss of orthogonality. Our bounds are derived in a general fashion to additionally allow for the analysis of mixed-precision variants. We verify our theoretical results with a panel of test matrices and experiments from a new version of the \texttt{BlockStab} toolbox.

We generalize McDiarmid's inequality for functions with bounded differences on a high probability set, using an extension argument. Those functions concentrate around their conditional expectations. We further extend the results to concentration in general metric spaces.