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We show that the decision problem of recognising whether a triangulated 3-manifold admits a Seifert fibered structure with non-empty boundary is in NP. We also show that the problem of producing Seifert data for a triangulation of such a manifold is in the complexity class FNP. We do this by proving that in any triangulation of a Seifert fibered space with boundary there is both a fundamental horizontal surface of small degree and a complete collection of normal vertical annuli whose total weight is bounded by an exponential in the square of the triangulation size.

The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.

The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm can be applied to stochastic differential equations to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method.

In this paper, practically computable low-order approximations of potentially high-dimensional differential equations driven by geometric rough paths are proposed and investigated. In particular, equations are studied that cover the linear setting, but we allow for a certain type of dissipative nonlinearity in the drift as well. In a first step, a linear subspace is found that contains the solution space of the underlying rough differential equation (RDE). This subspace is associated to covariances of linear Ito-stochastic differential equations which is shown exploiting a Gronwall lemma for matrix differential equations. Orthogonal projections onto the identified subspace lead to a first exact reduced order system. Secondly, a linear map of the RDE solution (quantity of interest) is analyzed in terms of redundant information meaning that state variables are found that do not contribute to the quantity of interest. Once more, a link to Ito-stochastic differential equations is used. Removing such unnecessary information from the RDE provides a further dimension reduction without causing an error. Finally, we discretize a linear parabolic rough partial differential equation in space. The resulting large-order RDE is subsequently tackled with the exact reduction techniques studied in this paper. We illustrate the enormous complexity reduction potential in the corresponding numerical experiments.

Deep neural networks often suffer from poor generalization due to complex and non-convex loss landscapes. Sharpness-Aware Minimization (SAM) is a popular solution that smooths the loss landscape by minimizing the maximized change of training loss when adding a perturbation to the weight. However, indiscriminate perturbation of SAM on all parameters is suboptimal and results in excessive computation, double the overhead of common optimizers like Stochastic Gradient Descent (SGD). In this paper, we propose Sparse SAM (SSAM), an efficient and effective training scheme that achieves sparse perturbation by a binary mask. To obtain the sparse mask, we provide two solutions based on Fisher information and dynamic sparse training, respectively. We investigate the impact of different masks, including unstructured, structured, and $N$:$M$ structured patterns, as well as explicit and implicit forms of implementing sparse perturbation. We theoretically prove that SSAM can converge at the same rate as SAM, i.e., $O(\log T/\sqrt{T})$. Sparse SAM has the potential to accelerate training and smooth the loss landscape effectively. Extensive experimental results on CIFAR and ImageNet-1K confirm that our method is superior to SAM in terms of efficiency, and the performance is preserved or even improved with a perturbation of merely 50\% sparsity. Code is available at //github.com/Mi-Peng/Systematic-Investigation-of-Sparse-Perturbed-Sharpness-Aware-Minimization-Optimizer.

We show that convex-concave Lipschitz stochastic saddle point problems (also known as stochastic minimax optimization) can be solved under the constraint of $(\epsilon,\delta)$-differential privacy with \emph{strong (primal-dual) gap} rate of $\tilde O\big(\frac{1}{\sqrt{n}} + \frac{\sqrt{d}}{n\epsilon}\big)$, where $n$ is the dataset size and $d$ is the dimension of the problem. This rate is nearly optimal, based on existing lower bounds in differentially private stochastic optimization. Specifically, we prove a tight upper bound on the strong gap via novel implementation and analysis of the recursive regularization technique repurposed for saddle point problems. We show that this rate can be attained with $O\big(\min\big\{\frac{n^2\epsilon^{1.5}}{\sqrt{d}}, n^{3/2}\big\}\big)$ gradient complexity, and $\tilde{O}(n)$ gradient complexity if the loss function is smooth. As a byproduct of our method, we develop a general algorithm that, given a black-box access to a subroutine satisfying a certain $\alpha$ primal-dual accuracy guarantee with respect to the empirical objective, gives a solution to the stochastic saddle point problem with a strong gap of $\tilde{O}(\alpha+\frac{1}{\sqrt{n}})$. We show that this $\alpha$-accuracy condition is satisfied by standard algorithms for the empirical saddle point problem such as the proximal point method and the stochastic gradient descent ascent algorithm. Further, we show that even for simple problems it is possible for an algorithm to have zero weak gap and suffer from $\Omega(1)$ strong gap. We also show that there exists a fundamental tradeoff between stability and accuracy. Specifically, we show that any $\Delta$-stable algorithm has empirical gap $\Omega\big(\frac{1}{\Delta n}\big)$, and that this bound is tight. This result also holds also more specifically for empirical risk minimization problems and may be of independent interest.

In this work, we consider a rational approximation of the exponential function to design an algorithm for computing matrix exponential in the Hermitian case. Using partial fraction decomposition, we obtain a parallelizable method, where the computation reduces to independent resolutions of linear systems. We analyze the effects of rounding errors on the accuracy of our algorithm. We complete this work with numerical tests showing the efficiency of our method and a comparison of its performances with Krylov algorithms.

Stochastic congestion, a phenomenon in which a system becomes temporarily overwhelmed by random surges in demand, occurs frequently in service applications. While randomized experiments have been effective in gaining causal insights and prescribing policy improvements in many domains, using them to study stochastic congestion has proven challenging. This is because congestion can induce interference between customers in the service system and thus hinder subsequent statistical analysis. In this paper, we aim at getting tailor-made experimental designs and estimators for the interference induced by stochastic congestion. In particular, taking a standard queueing system as a benchmark model of congestion, we study how to conduct randomized experiments in a service system that has a single queue with an outside option. We study switchback experiments and a local perturbation experiment and propose estimators based on the experiments to estimate the effect of a system parameter on the average arrival rate. We establish that the estimator from the local perturbation experiment is asymptotically more accurate than the estimators from the switchback experiments because it takes advantage of the structure of the queueing system.

Quantization is commonly used in Deep Neural Networks (DNNs) to reduce the storage and computational complexity by decreasing the arithmetical precision of activations and weights, a.k.a. tensors. Efficient hardware architectures employ linear quantization to enable the deployment of recent DNNs onto embedded systems and mobile devices. However, linear uniform quantization cannot usually reduce the numerical precision to less than 8 bits without sacrificing high performance in terms of model accuracy. The performance loss is due to the fact that tensors do not follow uniform distributions. In this paper, we show that a significant amount of tensors fit into an exponential distribution. Then, we propose DNA-TEQ to exponentially quantize DNN tensors with an adaptive scheme that achieves the best trade-off between numerical precision and accuracy loss. The experimental results show that DNA-TEQ provides a much lower quantization bit-width compared to previous proposals, resulting in an average compression ratio of 40% over the linear INT8 baseline, with negligible accuracy loss and without retraining the DNNs. Besides, DNA-TEQ leads the way in performing dot-product operations in the exponential domain, which saves 66% of energy consumption on average for a set of widely used DNNs.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.