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We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical systems. GCANs are based on symmetry group transformations using geometric (Clifford) algebras. We first review the quintessence of modern (plane-based) geometric algebra, which builds on isometries encoded as elements of the $\mathrm{Pin}(p,q,r)$ group. We then propose the concept of group action layers, which linearly combine object transformations using pre-specified group actions. Together with a new activation and normalization scheme, these layers serve as adjustable $\textit{geometric templates}$ that can be refined via gradient descent. Theoretical advantages are strongly reflected in the modeling of three-dimensional rigid body transformations as well as large-scale fluid dynamics simulations, showing significantly improved performance over traditional methods.

Using a perturbation approach, we derive a new approximate filtering and smoothing methodology for a general class of state-space models including univariate and multivariate location, scale, and count data models. The main properties of the methodology can be summarized as follows: (i) it generalizes several existing approaches to robust filtering based on the score and the Hessian matrix of the observation density by relaxing the critical assumption of a Gaussian prior density underlying this class of methods; (ii) has a very simple structure based on forward-backward recursions similar to the Kalman filter and smoother; (iii) allows a straightforward computation of confidence bands around the state estimates reflecting the combination of parameter and filtering uncertainty. We show through an extensive Monte Carlo study that the mean square loss with respect to exact simulation-based methods is small in a wide range of scenarios. We finally illustrate empirically the application of the methodology to the estimation of stochastic volatility and correlations in financial time-series.

Datasets with sheer volume have been generated from fields including computer vision, medical imageology, and astronomy whose large-scale and high-dimensional properties hamper the implementation of classical statistical models. To tackle the computational challenges, one of the efficient approaches is subsampling which draws subsamples from the original large datasets according to a carefully-design task-specific probability distribution to form an informative sketch. The computation cost is reduced by applying the original algorithm to the substantially smaller sketch. Previous studies associated with subsampling focused on non-regularized regression from the computational efficiency and theoretical guarantee perspectives, such as ordinary least square regression and logistic regression. In this article, we introduce a randomized algorithm under the subsampling scheme for the Elastic-net regression which gives novel insights into L1-norm regularized regression problem. To effectively conduct consistency analysis, a smooth approximation technique based on alpha absolute function is firstly employed and theoretically verified. The concentration bounds and asymptotic normality for the proposed randomized algorithm are then established under mild conditions. Moreover, an optimal subsampling probability is constructed according to A-optimality. The effectiveness of the proposed algorithm is demonstrated upon synthetic and real data datasets.

We consider repeated multi-unit auctions with uniform pricing, which are widely used in practice for allocating goods such as carbon licenses. In each round, $K$ identical units of a good are sold to a group of buyers that have valuations with diminishing marginal returns. The buyers submit bids for the units, and then a price $p$ is set per unit so that all the units are sold. We consider two variants of the auction, where the price is set to the $K$-th highest bid and $(K+1)$-st highest bid, respectively. We analyze the properties of this auction in both the offline and online settings. In the offline setting, we consider the problem that one player $i$ is facing: given access to a data set that contains the bids submitted by competitors in past auctions, find a bid vector that maximizes player $i$'s cumulative utility on the data set. We design a polynomial time algorithm for this problem, by showing it is equivalent to finding a maximum-weight path on a carefully constructed directed acyclic graph. In the online setting, the players run learning algorithms to update their bids as they participate in the auction over time. Based on our offline algorithm, we design efficient online learning algorithms for bidding. The algorithms have sublinear regret, under both full information and bandit feedback structures. We complement our online learning algorithms with regret lower bounds. Finally, we analyze the quality of the equilibria in the worst case through the lens of the core solution concept in the game among the bidders. We show that the $(K+1)$-st price format is susceptible to collusion among the bidders; meanwhile, the $K$-th price format does not have this issue.

We present a mass lumping approach based on an isogeometric Petrov-Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is ``close'' to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

Sliding window sums are widely used for string indexing, hashing and time series analysis. We have developed a family of the generic vectorized sliding sum algorithms that provide speedup of O(P/w) for window size $w$ and number of processors P. For a sum with a commutative operator the speedup is improved to O(P/log(w)). Even more important, our algorithms exhibit efficient memory access patterns. In this paper we study the application of the sliding sum algorithms to the training and inference of the Deep Neural Networks. We demonstrate how both pooling and convolution primitives could be expressed as sliding sums and evaluated by the compute kernels with the shared structure. We show that the sliding sum convolution kernels are more efficient than the commonly used GEMM kernels on the CPU, and could even outperform their GPU counterparts.

In this paper, we study the problem of optimal data collection for policy evaluation in linear bandits. In policy evaluation, we are given a target policy and asked to estimate the expected reward it will obtain when executed in a multi-armed bandit environment. Our work is the first work that focuses on such optimal data collection strategy for policy evaluation involving heteroscedastic reward noise in the linear bandit setting. We first formulate an optimal design for weighted least squares estimates in the heteroscedastic linear bandit setting that reduces the MSE of the value of the target policy. We then use this formulation to derive the optimal allocation of samples per action during data collection. We then introduce a novel algorithm SPEED (Structured Policy Evaluation Experimental Design) that tracks the optimal design and derive its regret with respect to the optimal design. Finally, we empirically validate that SPEED leads to policy evaluation with mean squared error comparable to the oracle strategy and significantly lower than simply running the target policy.

Recurrent neural networks are a powerful means to cope with time series. We show how autoregressive linear, i.e., linearly activated recurrent neural networks (LRNNs) can approximate any time-dependent function f(t) given by a number of function values. The approximation can effectively be learned by simply solving a linear equation system; no backpropagation or similar methods are needed. Furthermore, and this is probably the main contribution of this article, the size of an LRNN can be reduced significantly in one step after inspecting the spectrum of the network transition matrix, i.e., its eigenvalues, by taking only the most relevant components. Therefore, in contrast to other approaches, we do not only learn network weights but also the network architecture. LRNNs have interesting properties: They end up in ellipse trajectories in the long run and allow the prediction of further values and compact representations of functions. We demonstrate this by several experiments, among them multiple superimposed oscillators (MSO), robotic soccer, and predicting stock prices. LRNNs outperform the previous state-of-the-art for the MSO task with a minimal number of units.

DPPs were introduced by Macchi as a model in quantum optics the 1970s. Since then, they have been widely used as models and subsampling tools in statistics and computer science. Most applications require sampling from a DPP, and given their quantum origin, it is natural to wonder whether sampling a DPP on a quantum computer is easier than on a classical one. We focus here on DPPs over a finite state space, which are distributions over the subsets of $\{1,\dots,N\}$ parametrized by an $N\times N$ Hermitian kernel matrix. Vanilla sampling consists in two steps, of respective costs $\mathcal{O}(N^3)$ and $\mathcal{O}(Nr^2)$ operations on a classical computer, where $r$ is the rank of the kernel matrix. A large first part of the current paper consists in explaining why the state-of-the-art in quantum simulation of fermionic systems already yields quantum DPP sampling algorithms. We then modify existing quantum circuits, and discuss their insertion in a full DPP sampling pipeline that starts from practical kernel specifications. The bottom line is that, with $P$ (classical) parallel processors, we can divide the preprocessing cost by $P$ and build a quantum circuit with $\mathcal{O}(Nr)$ gates that sample a given DPP, with depth varying from $\mathcal{O}(N)$ to $\mathcal{O}(r\log N)$ depending on qubit-communication constraints on the target machine. We also connect existing work on the simulation of superconductors to Pfaffian point processes, which generalize DPPs and would be a natural addition to the machine learner's toolbox. Finally, the circuits are empirically validated on a classical simulator and on 5-qubit machines.

In this paper, we propose a new model for forecasting time series data distributed on a matrix-shaped spatial grid, using the historical spatio-temporal data together with auxiliary vector-valued time series data. We model the matrix time series as an auto-regressive process, where a future matrix is jointly predicted by the historical values of the matrix time series as well as an auxiliary vector time series. The matrix predictors are associated with row/column-specific autoregressive matrix coefficients that map the predictors to the future matrices via a bi-linear transformation. The vector predictors are mapped to matrices by taking mode product with a 3D coefficient tensor. Given the high dimensionality of the tensor coefficient and the underlying spatial structure of the data, we propose to estimate the tensor coefficient by estimating one functional coefficient for each covariate, with 2D input domain, from a Reproducing Kernel Hilbert Space. We jointly estimate the autoregressive matrix coefficients and the functional coefficients under a penalized maximum likelihood estimation framework, and couple it with an alternating minimization algorithm. Large sample asymptotics of the estimators are established and performances of the model are validated with extensive simulation studies and a real data application to forecast the global total electron content distributions.