We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some spacetime PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.
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To enable learning on edge devices with fast convergence and low memory, we present a novel backpropagation-free optimization algorithm dubbed Target Projection Stochastic Gradient Descent (tpSGD). tpSGD generalizes direct random target projection to work with arbitrary loss functions and extends target projection for training recurrent neural networks (RNNs) in addition to feedforward networks. tpSGD uses layer-wise stochastic gradient descent (SGD) and local targets generated via random projections of the labels to train the network layer-by-layer with only forward passes. tpSGD doesn't require retaining gradients during optimization, greatly reducing memory allocation compared to SGD backpropagation (BP) methods that require multiple instances of the entire neural network weights, input/output, and intermediate results. Our method performs comparably to BP gradient-descent within 5% accuracy on relatively shallow networks of fully connected layers, convolutional layers, and recurrent layers. tpSGD also outperforms other state-of-the-art gradient-free algorithms in shallow models consisting of multi-layer perceptrons, convolutional neural networks (CNNs), and RNNs with competitive accuracy and less memory and time. We evaluate the performance of tpSGD in training deep neural networks (e.g. VGG) and extend the approach to multi-layer RNNs. These experiments highlight new research directions related to optimized layer-based adaptor training for domain-shift using tpSGD at the edge.
This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function (Stein, 1999, Section 6.7) with Fourier coefficients $\phi$($\alpha$^2 + j^2)^(--$\nu$--1/2). Convergence rates are studied for the joint maximum likelihood estimation of $\nu$ and $\phi$ when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a ''deterministic'' element of a continuous Sobolev space is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.
Reconstructing 3D objects from 2D images is both challenging for our brains and machine learning algorithms. To support this spatial reasoning task, contextual information about the overall shape of an object is critical. However, such information is not captured by established loss terms (e.g. Dice loss). We propose to complement geometrical shape information by including multi-scale topological features, such as connected components, cycles, and voids, in the reconstruction loss. Our method uses cubical complexes to calculate topological features of 3D volume data and employs an optimal transport distance to guide the reconstruction process. This topology-aware loss is fully differentiable, computationally efficient, and can be added to any neural network. We demonstrate the utility of our loss by incorporating it into SHAPR, a model for predicting the 3D cell shape of individual cells based on 2D microscopy images. Using a hybrid loss that leverages both geometrical and topological information of single objects to assess their shape, we find that topological information substantially improves the quality of reconstructions, thus highlighting its ability to extract more relevant features from image datasets.
In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. A second-order scheme is first proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence in the $H^1$ norm without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to overcome the difficulty posed by the unavailability of a priori $L^\infty$ estimates of numerical solutions. Based on the integrating factor Runge-Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linear and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.
In this paper, we consider function-indexed normalized weighted integrated periodograms for equidistantly sampled multivariate continuous-time state space models which are multivariate continuous-time ARMA processes. Thereby, the sampling distance is fixed and the driving L\'evy process has at least a finite fourth moment. Under different assumptions on the function space and the moments of the driving L\'evy process we derive a central limit theorem for the function-indexed normalized weighted integrated periodogram. Either the assumption on the function space or the assumption on the existence of moments of the L\'evy process is weaker. Furthermore, we show the weak convergence in both the space of continuous functions and in the dual space to a Gaussian process and give an explicit representation of the covariance function. The results can be used to derive the asymptotic behavior of the Whittle estimator and to construct goodness-of-fit test statistics as the Grenander-Rosenblatt statistic and the Cram\'er-von Mises statistic. We present the exact limit distributions of both statistics and show their performance through a simulation study.
This thesis investigates the quality of randomly collected data by employing a framework built on information-based complexity, a field related to the numerical analysis of abstract problems. The quality or power of gathered information is measured by its radius which is the uniform error obtainable by the best possible algorithm using it. The main aim is to present progress towards understanding the power of random information for approximation and integration problems.
Efficiently reducing models of chemically reacting flows is often challenging because their characteristic features such as sharp gradients in the flow fields and couplings over various time and length scales lead to dynamics that evolve in high-dimensional spaces. In this work, we show that online adaptive reduced models that construct nonlinear approximations by adapting low-dimensional subspaces over time can predict well latent dynamics with properties similar to those found in chemically reacting flows. The adaptation of the subspaces is driven by the online adaptive empirical interpolation method, which takes sparse residual evaluations of the full model to compute low-rank basis updates of the subspaces. Numerical experiments with a premixed flame model problem show that reduced models based on online adaptive empirical interpolation accurately predict flame dynamics far outside of the training regime and in regimes where traditional static reduced models, which keep reduced spaces fixed over time and so provide only linear approximations of latent dynamics, fail to make meaningful predictions.
The Brazilian test has been extremely popular while prompting significant debate. The main source of controversy is rooted in its indirect nature; the material tensile strength is inferred upon assuming that cracking initiates at the centre of the sample. Here, we use the Griffith criterion and finite element analysis to map the conditions (jaws geometry and material properties) that result in the nucleation of a centre crack. Unlike previous studies, we do not restrict ourselves to evaluating the stress state at the disk centre; the failure envelope of the generalised Griffith criterion is used to establish the crack nucleation location. We find that the range of conditions where the Brazilian test is valid is much narrower than previously assumed, with current practices and standards being inappropriate for a wide range of rock-like materials. The results obtained are used to develop a protocol that experimentalists can follow to obtain a valid estimate of the material tensile strength. This is showcased with specific case studies and examples of valid and invalid tests from the literature. Furthermore, the uptake of this protocol is facilitated by providing a MATLAB App that determines the validity of the experiment for arbitrary test conditions.
Interpretability methods are developed to understand the working mechanisms of black-box models, which is crucial to their responsible deployment. Fulfilling this goal requires both that the explanations generated by these methods are correct and that people can easily and reliably understand them. While the former has been addressed in prior work, the latter is often overlooked, resulting in informal model understanding derived from a handful of local explanations. In this paper, we introduce explanation summary (ExSum), a mathematical framework for quantifying model understanding, and propose metrics for its quality assessment. On two domains, ExSum highlights various limitations in the current practice, helps develop accurate model understanding, and reveals easily overlooked properties of the model. We also connect understandability to other properties of explanations such as human alignment, robustness, and counterfactual minimality and plausibility.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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